# The Centres of a Triangle

> Draw a triangle as crooked as you like, apply one rule at each of its three corners, and the three lines you get refuse to miss each other. It happens four times over: the medians meet at the centroid, the balance point two thirds of the way down each one; the perpendicular bisectors meet at the circumcentre, the middle of the circle through all three corners; the angle bisectors meet at the incentre, the middle of the circle tucked inside; and the altitudes meet at the orthocentre, which walks straight out of the triangle when the triangle turns obtuse. Then three of those four points turn out to stand on one straight line.

- Canonical watch page: [The Centres of a Triangle](https://academa.ai/lectures/the-centres-of-a-triangle)
- Publisher: [Academa, Inc.](https://academa.ai)
- Subject: Mathematics
- Published: 2026-08-29T00:23:33.000Z
- Updated: 2026-08-29T00:23:33.000Z
- Duration: PT339S (5 minutes 39 seconds)
- Chapters: 3
- Views: 1
- Language: en-US
- Access: Free
- Video stream: [HLS content](https://academa.ai/media/l/01M14TZJ8RS7SN9K6VECDBV527/1/dark/master.m3u8)
- Audiovisual record: [Semantic JSON](https://academa.ai/media/l/01M14TZJ8RS7SN9K6VECDBV527/1/semantic.json)
- Thumbnail: [Image](https://academa.ai/media/l/01M14TZJ8RS7SN9K6VECDBV527/1/dark/poster.jpg)

## Description

Medians, perpendicular bisectors, angle bisectors and altitudes: four rules, four centres, and the straight line three of them share.

## Chapters

- [00:00–03:10.139 · Four Constructions](https://academa.ai/lectures/the-centres-of-a-triangle?t=0)
- [03:10.139–04:2.731 · When the Orthocentre Moves Outside](https://academa.ai/lectures/the-centres-of-a-triangle?t=190.1393333333333)
- [04:2.731–05:39 · The Euler Line](https://academa.ai/lectures/the-centres-of-a-triangle?t=242.73141666666663)

## Transcript

### [00:00 · Four Constructions](https://academa.ai/lectures/the-centres-of-a-triangle?t=0)

A triangle has more than corners and sides. Four different geometric rules locate four special points, and the surprise is that each rule makes three independently defined lines meet. The four destinations are the centroid G, the circumcentre O, the incentre I, and the orthocentre H. Their rules are different, so their dots occupy different places in the same triangle. The first rule begins at the midpoint of every side. Those three landing places are A prime, B prime, and C prime. A segment from a corner to the opposite midpoint is a median. The median from A lands at A prime, and the median from B lands at B prime. Those two meet. The third median, from C to C prime, passes through the very same point. That common point is the centroid G. The centroid is the triangle's balance point. Along the median from A, the length A G is two thirds of the whole median, while G A prime is one third. The two lengths therefore have ratio two to one. A second rule starts from the same midpoints, but turns through a right angle. A perpendicular bisector crosses a side square on and divides it into equal halves. Choose any point on that line. Its distance to B equals its distance to C, the two ends of the side it bisects. The other two perpendicular bisectors carry the same promise for their sides. All three pass through one point, the circumcentre O. Because O lies on all three bisectors, O A, O B, and O C are equal. Call their common length R. A circle with centre O and radius R therefore passes through A, B, and C. This is the circumcircle, and the gap from O back to G confirms that the circumcentre and centroid are different points. The third rule belongs to the corners. An angle bisector splits one corner into two equal angles. The angle bisectors from A and C meet the first at one point. This is the incentre I. The shortest distance from I to side A B is a perpendicular drop. The corresponding drops to B C and C A have exactly the same length. Their matching marks record the equality, and that common length is r. The circle centred at I with radius r reaches every side and crosses none. It touches A B, B C, and C A at the three perpendicular feet. This is the incircle. The fourth rule returns to perpendicular lines. An altitude passes through a corner and meets the opposite side at a right angle. The altitudes from B and C pass through the first altitude at one point. This fourth meeting point is the orthocentre H. For this acute triangle, every altitude foot and the orthocentre lie inside. An obtuse angle changes that geometry.

### [03:10.139 · When the Orthocentre Moves Outside](https://academa.ai/lectures/the-centres-of-a-triangle?t=190.1393333333333)

At R, the angle is greater than a right angle, so this triangle is obtuse. The question is still the same: where do its three altitudes meet? Continue side Q R beyond R. The perpendicular from P meets that extended line outside the triangle. Continue side R P beyond R as well. The perpendicular from Q has its own right-angle foot beyond the triangle. The altitude from R meets P Q on the side itself. Its right-angle mark is inside, but the line continues upward to meet the other two altitudes. All three altitudes still meet at one point. The orthocentre H lies outside an obtuse triangle. The defining rule did not fail. Extending the two short sides exposed the same concurrency beyond the triangle.

### [04:2.731 · The Euler Line](https://academa.ai/lectures/the-centres-of-a-triangle?t=242.73141666666663)

The four centres return on a triangle whose top corner can move. First, the three side midpoints mark where the medians land. The median from A reaches A prime, the median from B reaches B prime, and the median from C reaches C prime. Their meeting point is the centroid G. The perpendicular bisectors meet at the circumcentre O, the centre of the circle through all three corners. The altitudes meet at the orthocentre H. The construction lines can now leave. Only three labelled points remain: the circumcentre O, the centroid G, and the orthocentre H. They lie on one straight line, exactly. This is the Euler line. That alignment belongs to the triangle, not to this one shape. As the top corner moves left, O, G, and H move with it and remain collinear. Move the corner back to the right, and the same line carries all three centres again. Their spacing is fixed as well. O G is one part, while G H is two parts, so O G to G H is one to two. The incentre I also moves with the triangle, but it does not generally lie on the Euler line. The centroid G comes from medians, the circumcentre O from perpendicular bisectors, the incentre I from angle bisectors, and the orthocentre H from altitudes. Four independent rules reveal a remarkably organised triangle.

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## Complete audiovisual record

Immutable source: [semantic.json](https://academa.ai/media/l/01M14TZJ8RS7SN9K6VECDBV527/1/semantic.json)

Record version: 1. Render attempt: 1.

### How to read this timeline

Each scene owns its object identifiers. A beat's board is the complete board when listed, empty when marked empty, and unchanged from the nearest earlier listed board in the same scene when marked unchanged. Action times are absolute positions in the published video.

### Scene 1: [Four Constructions](https://academa.ai/lectures/the-centres-of-a-triangle?t=0)

Span: 00:00–03:10.139 (0s–190.1393333333333s).

#### Objects

- altitude\_a: a Line \[red\] drawn in plane (start=(7.08, 2.960000000000001), end=(1.0, 2.0))
- altitude\_b: a Line \[red\] drawn in plane (start=(5.2, 5.0), end=(7.2, 2.2))
- altitude\_c: a Line \[red\] drawn in plane (start=(6.723076923076922, 2.184615384615385), end=(6.6, 6.0))
- altitude\_note: a Panel that says "An altitude passes through a corner and meets the opposite side at a right angle. It is the height measured from that corner."
- altitude\_square\_a: an Angle \[red\] drawn in plane (vertex=(7.08, 2.960000000000001), sides=((6.6, 6.0), (1.0, 2.0)), right\_angle=True)
- altitude\_square\_b: an Angle \[red\] drawn in plane (vertex=(5.2, 5.0), sides=((1.0, 2.0), (7.2, 2.2)), right\_angle=True)
- altitude\_square\_c: an Angle \[red\] drawn in plane (vertex=(6.723076923076922, 2.184615384615385), sides=((7.2, 2.2), (6.6, 6.0)), right\_angle=True)
- angle\_bisector\_a: a Line \[magenta\] drawn in plane (start=(1.0, 2.0), end=(6.915558979628808, 4.001459795684212))
- angle\_bisector\_b: a Line \[magenta\] drawn in plane (start=(7.2, 2.2), end=(4.456419576191015, 4.468871125850725))
- angle\_bisector\_c: a Line \[magenta\] drawn in plane (start=(6.6, 6.0), end=(4.97686492345502, 2.1282859652727426))
- angle\_note: a Panel that says "An angle bisector splits a corner into two equal angles. Every point on it is equally far from the angle's two sides."
- bisector\_a: a Line \[gray\] drawn in plane (start=(7.196328889598721, 4.146788772041903), end=(3.640382214414073, 3.5853235075390635))
- bisector\_b: a Line \[gray\] drawn in plane (start=(3.625628541884271, 4.24412004136202), end=(5.718086039273018, 1.3146795450177744))
- bisector\_c: a Line \[gray\] drawn in plane (start=(4.109672388203287, 1.80015596569809), end=(3.993603729763838, 5.3982843773210085))
- card: a Title that says "Plane Geometry — The Centres of a Triangle"
- centre\_g: a Point \[green\] labelled "G" drawn in plane (location=(4.933333333333333, 3.4))
- centre\_h: a Point \[red\] labelled "H" drawn in plane (location=(6.699999999999999, 2.900000000000001))
- centre\_i: a Point \[magenta\] labelled "I" drawn in plane (location=(5.57151267349114, 3.5467175380994163))
- centre\_o: a Point \[yellow\] labelled "O" drawn in plane (location=(4.050000000000001, 3.6499999999999995))
- circumcircle: a Circle \[yellow\] drawn in plane (center=(4.050000000000001, 3.650000000000001), radius=3.4677081768799414)
- destination\_note: a Panel that says "Medians meet at $G$, perpendicular bisectors at $O$, angle bisectors at $I$, and altitudes at $H$."
- distance\_formula: a Math \[text\] that says "$d(I, A B) = d(I, B C) = d(I, C A) = r$"
- drop\_a: a Line \[yellow\] drawn in plane (start=(6.9529208497020365, 3.764834618553768), end=(5.57151267349114, 3.5467175380994163), dashed=True)
- drop\_b: a Line \[yellow\] drawn in plane (start=(4.758638308574938, 4.684741648982099), end=(5.57151267349114, 3.5467175380994163), dashed=True)
- drop\_c: a Line \[yellow\] drawn in plane (start=(5.6166028304636875, 2.1489226719504417), end=(5.57151267349114, 3.5467175380994163), dashed=True)
- half\_b1: an Angle \[magenta\] drawn in plane (vertex=(7.2, 2.2), sides=((1.0, 2.0), (4.456419576191015, 4.468871125850725)), radius=0.65)
- half\_b2: an Angle \[magenta\] drawn in plane (vertex=(7.2, 2.2), sides=((4.456419576191015, 4.468871125850725), (6.6, 6.0)), radius=0.65)
- heading\_centres: a Heading that says "Four Rules, Four Centres"
- heading\_g: a Heading that says "The Centroid"
- heading\_h: a Heading that says "The Orthocentre"
- heading\_i: a Heading that says "The Incentre"
- heading\_o: a Heading that says "The Circumcentre"
- incircle: a Circle \[magenta\] drawn in plane (center=(5.57151267349114, 3.5467175380994163), radius=1.398521937649975)
- median\_a: a Line \[green\] drawn in plane (start=(1.0, 2.0), end=(6.9, 4.1))
- median\_b: a Line \[green\] drawn in plane (start=(7.2, 2.2), end=(3.8, 4.0))
- median\_c: a Line \[green\] drawn in plane (start=(6.6, 6.0), end=(4.1, 2.1))
- median\_note: a Panel that says "A median joins a corner to the midpoint of the opposite side. Every triangle has three."
- mid\_a: a Point \[text\] labelled "A'" drawn in plane (location=(6.9, 4.1))
- mid\_b: a Point \[text\] labelled "B'" drawn in plane (location=(3.8, 4.0))
- mid\_c: a Point \[text\] labelled "C'" drawn in plane (location=(4.1, 2.1))
- perpendicular\_note: a Panel that says "A perpendicular bisector crosses a side at its midpoint and at a right angle. Every point on it is equally far from the side's ends."
- plane: a Figure (x\_range=(0.2, 7.9), y\_range=(0.0, 7.35), aspect=(7.7, 7.35))
- point: a Point \[yellow\] drawn in plane (location=(4.933333333333333, 3.4))
- point\_2: a Point \[yellow\] drawn in plane (location=(4.991641950984239, 3.798680308050143))
- point\_3: a Point \[yellow\] drawn in plane (location=(5.6166028304636875, 2.1489226719504417))
- point\_4: a Point \[yellow\] drawn in plane (location=(6.9529208497020365, 3.764834618553768))
- point\_5: a Point \[yellow\] drawn in plane (location=(4.758638308574938, 4.684741648982099))
- point\_6: a Point \[yellow\] drawn in plane (location=(7.08, 2.960000000000001))
- point\_7: a Point \[yellow\] drawn in plane (location=(5.2, 5.0))
- point\_8: a Point \[yellow\] drawn in plane (location=(6.723076923076922, 2.184615384615385))
- radius\_formula: a Math \[text\] that says "$O A = O B = O C = R$"
- ratio\_g: a Math \[text\] that says "$A G : G A' = 2 : 1$"
- square\_a: an Angle \[gray\] drawn in plane (vertex=(6.9, 4.1), sides=((6.6, 6.0), (3.640382214414073, 3.5853235075390635)), right\_angle=True)
- square\_b: an Angle \[gray\] drawn in plane (vertex=(3.8, 4.0), sides=((1.0, 2.0), (5.718086039273018, 1.3146795450177744)), right\_angle=True)
- square\_c: an Angle \[gray\] drawn in plane (vertex=(4.1, 2.1), sides=((7.2, 2.2), (3.993603729763838, 5.3982843773210085)), right\_angle=True)
- triangle: a Polygon \[blue\] drawn in plane (vertices=((1.0, 2.0), (7.2, 2.2), (6.6, 6.0)), fill\_opacity=0.12)
- vertex\_a: a Point \[text\] labelled "A" drawn in plane (location=(1.0, 2.0))
- vertex\_b: a Point \[text\] labelled "B" drawn in plane (location=(7.2, 2.2))
- vertex\_c: a Point \[text\] labelled "C" drawn in plane (location=(6.6, 6.0))

#### Beats

##### [00:00](https://academa.ai/lectures/the-centres-of-a-triangle?t=0)

Narration: A triangle has more than corners and sides. Four different geometric rules locate four special points, and the surprise is that each rule makes three independently defined lines meet.

Board: Empty.

Actions:
- [00:00](https://academa.ai/lectures/the-centres-of-a-triangle?t=0): card is shown on the screen, written out.
- [00:1.5](https://academa.ai/lectures/the-centres-of-a-triangle?t=1.5): card: enter:write-left-to-right.
- [00:11.494](https://academa.ai/lectures/the-centres-of-a-triangle?t=11.494): card is hidden from the screen — left the board.

##### [00:12.694](https://academa.ai/lectures/the-centres-of-a-triangle?t=12.693999999999999)

Narration: The four destinations are the centroid G, the circumcentre O, the incentre I, and the orthocentre H. Their rules are different, so their dots occupy different places in the same triangle.

Board: Unchanged from the preceding beat in this scene.

Actions:
- [00:12.694](https://academa.ai/lectures/the-centres-of-a-triangle?t=12.693999999999999): heading\_centres is shown on the screen, written out.
- [00:12.694](https://academa.ai/lectures/the-centres-of-a-triangle?t=12.693999999999999): plane is shown on the screen, written out.
- [00:12.694](https://academa.ai/lectures/the-centres-of-a-triangle?t=12.693999999999999): triangle is shown on the screen, written out.
- [00:12.694](https://academa.ai/lectures/the-centres-of-a-triangle?t=12.693999999999999): vertex\_a is shown on the screen, written out.
- [00:12.694](https://academa.ai/lectures/the-centres-of-a-triangle?t=12.693999999999999): vertex\_b is shown on the screen, written out.
- [00:12.694](https://academa.ai/lectures/the-centres-of-a-triangle?t=12.693999999999999): vertex\_c is shown on the screen, written out.
- [00:13.518](https://academa.ai/lectures/the-centres-of-a-triangle?t=13.517999999999999): destination\_note is shown on the screen, written out.
- [00:14.958](https://academa.ai/lectures/the-centres-of-a-triangle?t=14.957999999999998): centre\_g is shown on the screen, written out.
- [00:14.958](https://academa.ai/lectures/the-centres-of-a-triangle?t=14.957999999999998): centre\_g is indicated — a transient flash.
- [00:16.572](https://academa.ai/lectures/the-centres-of-a-triangle?t=16.572): centre\_o is shown on the screen, written out.
- [00:16.572](https://academa.ai/lectures/the-centres-of-a-triangle?t=16.572): centre\_o is indicated — a transient flash.
- [00:18.035](https://academa.ai/lectures/the-centres-of-a-triangle?t=18.035): centre\_i is shown on the screen, written out.
- [00:18.035](https://academa.ai/lectures/the-centres-of-a-triangle?t=18.035): centre\_i is indicated — a transient flash.
- [00:19.765](https://academa.ai/lectures/the-centres-of-a-triangle?t=19.765): centre\_h is shown on the screen, written out.
- [00:19.765](https://academa.ai/lectures/the-centres-of-a-triangle?t=19.765): centre\_h is indicated — a transient flash.

##### [00:26.413](https://academa.ai/lectures/the-centres-of-a-triangle?t=26.4135)

Narration: The first rule begins at the midpoint of every side. Those three landing places are A prime, B prime, and C prime.

Board: destination\_note — a Panel that says "Medians meet at $G$, perpendicular bisectors at $O$, angle bisectors at $I$, and altitudes at $H$."; plane — a Figure (x\_range=(0.2, 7.9), y\_range=(0.0, 7.35), aspect=(7.7, 7.35)); heading\_centres — a Heading that says "Four Rules, Four Centres"; triangle — a Polygon \[blue\] drawn in plane (vertices=((1.0, 2.0), (7.2, 2.2), (6.6, 6.0)), fill\_opacity=0.12); vertex\_a — a Point \[text\] labelled "A" drawn in plane (location=(1.0, 2.0)); vertex\_b — a Point \[text\] labelled "B" drawn in plane (location=(7.2, 2.2)); vertex\_c — a Point \[text\] labelled "C" drawn in plane (location=(6.6, 6.0)); centre\_g — a Point \[green\] labelled "G" drawn in plane (location=(4.933333333333333, 3.4)); centre\_o — a Point \[yellow\] labelled "O" drawn in plane (location=(4.050000000000001, 3.6499999999999995)); centre\_i — a Point \[magenta\] labelled "I" drawn in plane (location=(5.57151267349114, 3.5467175380994163)); centre\_h — a Point \[red\] labelled "H" drawn in plane (location=(6.699999999999999, 2.900000000000001))

Actions:
- [00:26.413](https://academa.ai/lectures/the-centres-of-a-triangle?t=26.4135): centre\_g is hidden from the screen.
- [00:26.413](https://academa.ai/lectures/the-centres-of-a-triangle?t=26.4135): centre\_o is hidden from the screen.
- [00:26.413](https://academa.ai/lectures/the-centres-of-a-triangle?t=26.4135): centre\_i is hidden from the screen.
- [00:26.413](https://academa.ai/lectures/the-centres-of-a-triangle?t=26.4135): centre\_h is hidden from the screen.
- [00:31.545](https://academa.ai/lectures/the-centres-of-a-triangle?t=31.545): mid\_a is shown on the screen, written out.
- [00:32.288](https://academa.ai/lectures/the-centres-of-a-triangle?t=32.288000000000004): mid\_b is shown on the screen, written out.
- [00:33.438](https://academa.ai/lectures/the-centres-of-a-triangle?t=33.438): mid\_c is shown on the screen, written out.
- [00:34.505](https://academa.ai/lectures/the-centres-of-a-triangle?t=34.5055): destination\_note is hidden from the screen — left the board.
- [00:34.505](https://academa.ai/lectures/the-centres-of-a-triangle?t=34.5055): heading\_centres is hidden from the screen — left the board.

##### [00:35.105](https://academa.ai/lectures/the-centres-of-a-triangle?t=35.1055)

Narration: A segment from a corner to the opposite midpoint is a median. The median from A lands at A prime, and the median from B lands at B prime.

Board: plane — a Figure (x\_range=(0.2, 7.9), y\_range=(0.0, 7.35), aspect=(7.7, 7.35)); triangle — a Polygon \[blue\] drawn in plane (vertices=((1.0, 2.0), (7.2, 2.2), (6.6, 6.0)), fill\_opacity=0.12); vertex\_a — a Point \[text\] labelled "A" drawn in plane (location=(1.0, 2.0)); vertex\_b — a Point \[text\] labelled "B" drawn in plane (location=(7.2, 2.2)); vertex\_c — a Point \[text\] labelled "C" drawn in plane (location=(6.6, 6.0)); mid\_a — a Point \[text\] labelled "A'" drawn in plane (location=(6.9, 4.1)); mid\_b — a Point \[text\] labelled "B'" drawn in plane (location=(3.8, 4.0)); mid\_c — a Point \[text\] labelled "C'" drawn in plane (location=(4.1, 2.1))

Actions:
- [00:35.105](https://academa.ai/lectures/the-centres-of-a-triangle?t=35.1055): heading\_g is shown on the screen, written out.
- [00:35.105](https://academa.ai/lectures/the-centres-of-a-triangle?t=35.1055): median\_note is shown on the screen, written out.
- [00:36.034](https://academa.ai/lectures/the-centres-of-a-triangle?t=36.034): median\_a is shown on the screen, drawn.
- [00:38.055](https://academa.ai/lectures/the-centres-of-a-triangle?t=38.055): median\_note (the "median" part) is emphasized.
- [00:42.315](https://academa.ai/lectures/the-centres-of-a-triangle?t=42.315): median\_b is shown on the screen, drawn.

##### [00:44.831](https://academa.ai/lectures/the-centres-of-a-triangle?t=44.831)

Narration: Those two meet. The third median, from C to C prime, passes through the very same point. That common point is the centroid G.

Board: plane — a Figure (x\_range=(0.2, 7.9), y\_range=(0.0, 7.35), aspect=(7.7, 7.35)); triangle — a Polygon \[blue\] drawn in plane (vertices=((1.0, 2.0), (7.2, 2.2), (6.6, 6.0)), fill\_opacity=0.12); vertex\_a — a Point \[text\] labelled "A" drawn in plane (location=(1.0, 2.0)); vertex\_b — a Point \[text\] labelled "B" drawn in plane (location=(7.2, 2.2)); vertex\_c — a Point \[text\] labelled "C" drawn in plane (location=(6.6, 6.0)); mid\_a — a Point \[text\] labelled "A'" drawn in plane (location=(6.9, 4.1)); mid\_b — a Point \[text\] labelled "B'" drawn in plane (location=(3.8, 4.0)); mid\_c — a Point \[text\] labelled "C'" drawn in plane (location=(4.1, 2.1)); median\_note — a Panel that says "A median joins a corner to the midpoint of the opposite side. Every triangle has three."; heading\_g — a Heading that says "The Centroid"; median\_a — a Line \[green\] drawn in plane (start=(1.0, 2.0), end=(6.9, 4.1)); median\_b — a Line \[green\] drawn in plane (start=(7.2, 2.2), end=(3.8, 4.0))

Actions:
- [00:44.831](https://academa.ai/lectures/the-centres-of-a-triangle?t=44.831): median\_note (the "median" part) is no longer emphasized.
- [00:46.932](https://academa.ai/lectures/the-centres-of-a-triangle?t=46.932): median\_c is shown on the screen, drawn.
- [00:50.729](https://academa.ai/lectures/the-centres-of-a-triangle?t=50.729): centre\_g is shown on the screen, written out.
- [00:53.341](https://academa.ai/lectures/the-centres-of-a-triangle?t=53.341): centre\_g is indicated — a transient flash.

##### [00:55.044](https://academa.ai/lectures/the-centres-of-a-triangle?t=55.0445)

Narration: The centroid is the triangle's balance point. Along the median from A, the length A G is two thirds of the whole median, while G A prime is one third. The two lengths therefore have ratio two to one.

Board: plane — a Figure (x\_range=(0.2, 7.9), y\_range=(0.0, 7.35), aspect=(7.7, 7.35)); triangle — a Polygon \[blue\] drawn in plane (vertices=((1.0, 2.0), (7.2, 2.2), (6.6, 6.0)), fill\_opacity=0.12); vertex\_a — a Point \[text\] labelled "A" drawn in plane (location=(1.0, 2.0)); vertex\_b — a Point \[text\] labelled "B" drawn in plane (location=(7.2, 2.2)); vertex\_c — a Point \[text\] labelled "C" drawn in plane (location=(6.6, 6.0)); centre\_g — a Point \[green\] labelled "G" drawn in plane (location=(4.933333333333333, 3.4)); mid\_a — a Point \[text\] labelled "A'" drawn in plane (location=(6.9, 4.1)); mid\_b — a Point \[text\] labelled "B'" drawn in plane (location=(3.8, 4.0)); mid\_c — a Point \[text\] labelled "C'" drawn in plane (location=(4.1, 2.1)); median\_note — a Panel that says "A median joins a corner to the midpoint of the opposite side. Every triangle has three."; heading\_g — a Heading that says "The Centroid"; median\_a — a Line \[green\] drawn in plane (start=(1.0, 2.0), end=(6.9, 4.1)); median\_b — a Line \[green\] drawn in plane (start=(7.2, 2.2), end=(3.8, 4.0)); median\_c — a Line \[green\] drawn in plane (start=(6.6, 6.0), end=(4.1, 2.1))

Actions:
- [00:56.669](https://academa.ai/lectures/the-centres-of-a-triangle?t=56.669000000000004): point is shown on the screen, grown.
- [00:58.169](https://academa.ai/lectures/the-centres-of-a-triangle?t=58.169000000000004): point is hidden from the screen.
- [01:0.466](https://academa.ai/lectures/the-centres-of-a-triangle?t=60.466): ratio\_g is shown on the screen, written out.
- [01:0.466](https://academa.ai/lectures/the-centres-of-a-triangle?t=60.466): ratio\_g (the "A G" part) is emphasized.
- [01:0.466](https://academa.ai/lectures/the-centres-of-a-triangle?t=60.466): The segment (1.0, 2.0) to (4.933333333333333, 3.4) in plane is lit up.
- [01:1.197](https://academa.ai/lectures/the-centres-of-a-triangle?t=61.197): ratio\_g (the "2" part) is emphasized.
- [01:1.197](https://academa.ai/lectures/the-centres-of-a-triangle?t=61.197): ratio\_g (the "A G" part) is no longer emphasized.
- [01:3.148](https://academa.ai/lectures/the-centres-of-a-triangle?t=63.148): ratio\_g (the "2" part) is no longer emphasized.
- [01:3.148](https://academa.ai/lectures/the-centres-of-a-triangle?t=63.148): ratio\_g (the "G A'" part) is emphasized.
- [01:3.148](https://academa.ai/lectures/the-centres-of-a-triangle?t=63.148): plane: retire a lit segment (unemphasize\_line).
- [01:3.148](https://academa.ai/lectures/the-centres-of-a-triangle?t=63.148): The segment (4.933333333333333, 3.4) to (6.9, 4.1) in plane is lit up.
- [01:4.239](https://academa.ai/lectures/the-centres-of-a-triangle?t=64.239): ratio\_g (the "1" part) is emphasized.
- [01:4.239](https://academa.ai/lectures/the-centres-of-a-triangle?t=64.239): ratio\_g (the "G A'" part) is no longer emphasized.
- [01:8.837](https://academa.ai/lectures/the-centres-of-a-triangle?t=68.8365): heading\_g is hidden from the screen — left the board.
- [01:8.837](https://academa.ai/lectures/the-centres-of-a-triangle?t=68.8365): median\_note is hidden from the screen — left the board.
- [01:8.837](https://academa.ai/lectures/the-centres-of-a-triangle?t=68.8365): ratio\_g is hidden from the screen — left the board.
- [01:8.837](https://academa.ai/lectures/the-centres-of-a-triangle?t=68.8365): ratio\_g (the "1" part) is no longer emphasized.
- [01:8.837](https://academa.ai/lectures/the-centres-of-a-triangle?t=68.8365): plane: retire a lit segment (unemphasize\_line).

##### [01:9.436](https://academa.ai/lectures/the-centres-of-a-triangle?t=69.4365)

Narration: A second rule starts from the same midpoints, but turns through a right angle. A perpendicular bisector crosses a side square on and divides it into equal halves.

Board: plane — a Figure (x\_range=(0.2, 7.9), y\_range=(0.0, 7.35), aspect=(7.7, 7.35)); triangle — a Polygon \[blue\] drawn in plane (vertices=((1.0, 2.0), (7.2, 2.2), (6.6, 6.0)), fill\_opacity=0.12); vertex\_a — a Point \[text\] labelled "A" drawn in plane (location=(1.0, 2.0)); vertex\_b — a Point \[text\] labelled "B" drawn in plane (location=(7.2, 2.2)); vertex\_c — a Point \[text\] labelled "C" drawn in plane (location=(6.6, 6.0)); centre\_g — a Point \[green\] labelled "G" drawn in plane (location=(4.933333333333333, 3.4)); mid\_a — a Point \[text\] labelled "A'" drawn in plane (location=(6.9, 4.1)); mid\_b — a Point \[text\] labelled "B'" drawn in plane (location=(3.8, 4.0)); mid\_c — a Point \[text\] labelled "C'" drawn in plane (location=(4.1, 2.1)); median\_a — a Line \[green\] drawn in plane (start=(1.0, 2.0), end=(6.9, 4.1)); median\_b — a Line \[green\] drawn in plane (start=(7.2, 2.2), end=(3.8, 4.0)); median\_c — a Line \[green\] drawn in plane (start=(6.6, 6.0), end=(4.1, 2.1))

Actions:
- [01:9.436](https://academa.ai/lectures/the-centres-of-a-triangle?t=69.4365): median\_a is hidden from the screen.
- [01:9.436](https://academa.ai/lectures/the-centres-of-a-triangle?t=69.4365): median\_b is hidden from the screen.
- [01:9.436](https://academa.ai/lectures/the-centres-of-a-triangle?t=69.4365): median\_c is hidden from the screen.
- [01:9.436](https://academa.ai/lectures/the-centres-of-a-triangle?t=69.4365): centre\_g is hidden from the screen.
- [01:9.436](https://academa.ai/lectures/the-centres-of-a-triangle?t=69.4365): heading\_o is shown on the screen, written out.
- [01:10.319](https://academa.ai/lectures/the-centres-of-a-triangle?t=70.31899999999999): perpendicular\_note is shown on the screen, written out.
- [01:10.598](https://academa.ai/lectures/the-centres-of-a-triangle?t=70.598): bisector\_a is shown on the screen, drawn.
- [01:13.14](https://academa.ai/lectures/the-centres-of-a-triangle?t=73.13999999999999): square\_a is shown on the screen, written out.
- [01:13.14](https://academa.ai/lectures/the-centres-of-a-triangle?t=73.13999999999999): perpendicular\_note (the "right angle" part) is emphasized.
- [01:15.544](https://academa.ai/lectures/the-centres-of-a-triangle?t=75.544): perpendicular\_note (the "perpendicular bisector" part) is emphasized.
- [01:15.544](https://academa.ai/lectures/the-centres-of-a-triangle?t=75.544): perpendicular\_note (the "right angle" part) is no longer emphasized.

##### [01:20.497](https://academa.ai/lectures/the-centres-of-a-triangle?t=80.497)

Narration: Choose any point on that line. Its distance to B equals its distance to C, the two ends of the side it bisects.

Board: plane — a Figure (x\_range=(0.2, 7.9), y\_range=(0.0, 7.35), aspect=(7.7, 7.35)); triangle — a Polygon \[blue\] drawn in plane (vertices=((1.0, 2.0), (7.2, 2.2), (6.6, 6.0)), fill\_opacity=0.12); vertex\_a — a Point \[text\] labelled "A" drawn in plane (location=(1.0, 2.0)); vertex\_b — a Point \[text\] labelled "B" drawn in plane (location=(7.2, 2.2)); vertex\_c — a Point \[text\] labelled "C" drawn in plane (location=(6.6, 6.0)); mid\_a — a Point \[text\] labelled "A'" drawn in plane (location=(6.9, 4.1)); mid\_b — a Point \[text\] labelled "B'" drawn in plane (location=(3.8, 4.0)); mid\_c — a Point \[text\] labelled "C'" drawn in plane (location=(4.1, 2.1)); perpendicular\_note — a Panel that says "A perpendicular bisector crosses a side at its midpoint and at a right angle. Every point on it is equally far from the side's ends."; heading\_o — a Heading that says "The Circumcentre"; bisector\_a — a Line \[gray\] drawn in plane (start=(7.196328889598721, 4.146788772041903), end=(3.640382214414073, 3.5853235075390635)); square\_a — an Angle \[gray\] drawn in plane (vertex=(6.9, 4.1), sides=((6.6, 6.0), (3.640382214414073, 3.5853235075390635)), right\_angle=True)

Actions:
- [01:20.497](https://academa.ai/lectures/the-centres-of-a-triangle?t=80.497): perpendicular\_note (the "perpendicular bisector" part) is no longer emphasized.
- [01:21.449](https://academa.ai/lectures/the-centres-of-a-triangle?t=81.449): point\_2 is shown on the screen, grown.
- [01:23.449](https://academa.ai/lectures/the-centres-of-a-triangle?t=83.449): point\_2 is hidden from the screen.
- [01:23.98](https://academa.ai/lectures/the-centres-of-a-triangle?t=83.98): The segment (4.991641950984239, 3.798680308050143) to (7.2, 2.2) in plane is lit up.
- [01:25.42](https://academa.ai/lectures/the-centres-of-a-triangle?t=85.42): The segment (4.991641950984239, 3.798680308050143) to (6.6, 6.0) in plane is lit up.
- [01:25.42](https://academa.ai/lectures/the-centres-of-a-triangle?t=85.42): plane: retire a lit segment (unemphasize\_line).
- [01:28.392](https://academa.ai/lectures/the-centres-of-a-triangle?t=88.392): plane: retire a lit segment (unemphasize\_line).

##### [01:28.992](https://academa.ai/lectures/the-centres-of-a-triangle?t=88.992)

Narration: The other two perpendicular bisectors carry the same promise for their sides. All three pass through one point, the circumcentre O.

Board: Unchanged from the preceding beat in this scene.

Actions:
- [01:29.503](https://academa.ai/lectures/the-centres-of-a-triangle?t=89.50300000000001): bisector\_b is shown on the screen, drawn.
- [01:29.503](https://academa.ai/lectures/the-centres-of-a-triangle?t=89.50300000000001): square\_b is shown on the screen, written out.
- [01:29.712](https://academa.ai/lectures/the-centres-of-a-triangle?t=89.712): bisector\_c is shown on the screen, drawn.
- [01:29.712](https://academa.ai/lectures/the-centres-of-a-triangle?t=89.712): square\_c is shown on the screen, written out.
- [01:35.9](https://academa.ai/lectures/the-centres-of-a-triangle?t=95.9): centre\_o is shown on the screen, written out.
- [01:36.608](https://academa.ai/lectures/the-centres-of-a-triangle?t=96.608): centre\_o is indicated — a transient flash.

##### [01:37.782](https://academa.ai/lectures/the-centres-of-a-triangle?t=97.7825)

Narration: Because O lies on all three bisectors, O A, O B, and O C are equal. Call their common length R.

Board: plane — a Figure (x\_range=(0.2, 7.9), y\_range=(0.0, 7.35), aspect=(7.7, 7.35)); triangle — a Polygon \[blue\] drawn in plane (vertices=((1.0, 2.0), (7.2, 2.2), (6.6, 6.0)), fill\_opacity=0.12); vertex\_a — a Point \[text\] labelled "A" drawn in plane (location=(1.0, 2.0)); vertex\_b — a Point \[text\] labelled "B" drawn in plane (location=(7.2, 2.2)); vertex\_c — a Point \[text\] labelled "C" drawn in plane (location=(6.6, 6.0)); centre\_o — a Point \[yellow\] labelled "O" drawn in plane (location=(4.050000000000001, 3.6499999999999995)); mid\_a — a Point \[text\] labelled "A'" drawn in plane (location=(6.9, 4.1)); mid\_b — a Point \[text\] labelled "B'" drawn in plane (location=(3.8, 4.0)); mid\_c — a Point \[text\] labelled "C'" drawn in plane (location=(4.1, 2.1)); perpendicular\_note — a Panel that says "A perpendicular bisector crosses a side at its midpoint and at a right angle. Every point on it is equally far from the side's ends."; heading\_o — a Heading that says "The Circumcentre"; bisector\_a — a Line \[gray\] drawn in plane (start=(7.196328889598721, 4.146788772041903), end=(3.640382214414073, 3.5853235075390635)); square\_a — an Angle \[gray\] drawn in plane (vertex=(6.9, 4.1), sides=((6.6, 6.0), (3.640382214414073, 3.5853235075390635)), right\_angle=True); bisector\_b — a Line \[gray\] drawn in plane (start=(3.625628541884271, 4.24412004136202), end=(5.718086039273018, 1.3146795450177744)); square\_b — an Angle \[gray\] drawn in plane (vertex=(3.8, 4.0), sides=((1.0, 2.0), (5.718086039273018, 1.3146795450177744)), right\_angle=True); bisector\_c — a Line \[gray\] drawn in plane (start=(4.109672388203287, 1.80015596569809), end=(3.993603729763838, 5.3982843773210085)); square\_c — an Angle \[gray\] drawn in plane (vertex=(4.1, 2.1), sides=((7.2, 2.2), (3.993603729763838, 5.3982843773210085)), right\_angle=True)

Actions:
- [01:40.853](https://academa.ai/lectures/the-centres-of-a-triangle?t=100.85300000000001): radius\_formula is shown on the screen, written out.
- [01:40.853](https://academa.ai/lectures/the-centres-of-a-triangle?t=100.85300000000001): radius\_formula (the "O A" part) is emphasized.
- [01:40.853](https://academa.ai/lectures/the-centres-of-a-triangle?t=100.85300000000001): The segment (4.050000000000001, 3.6499999999999995) to (1.0, 2.0) in plane is lit up.
- [01:41.689](https://academa.ai/lectures/the-centres-of-a-triangle?t=101.68900000000001): radius\_formula (the "O A" part) is no longer emphasized.
- [01:41.689](https://academa.ai/lectures/the-centres-of-a-triangle?t=101.68900000000001): radius\_formula (the "O B" part) is emphasized.
- [01:41.689](https://academa.ai/lectures/the-centres-of-a-triangle?t=101.68900000000001): plane: retire a lit segment (unemphasize\_line).
- [01:41.689](https://academa.ai/lectures/the-centres-of-a-triangle?t=101.68900000000001): The segment (4.050000000000001, 3.6499999999999995) to (7.2, 2.2) in plane is lit up.
- [01:42.641](https://academa.ai/lectures/the-centres-of-a-triangle?t=102.641): radius\_formula (the "O B" part) is no longer emphasized.
- [01:42.641](https://academa.ai/lectures/the-centres-of-a-triangle?t=102.641): radius\_formula (the "O C" part) is emphasized.
- [01:42.641](https://academa.ai/lectures/the-centres-of-a-triangle?t=102.641): The segment (4.050000000000001, 3.6499999999999995) to (6.6, 6.0) in plane is lit up.
- [01:42.641](https://academa.ai/lectures/the-centres-of-a-triangle?t=102.641): plane: retire a lit segment (unemphasize\_line).
- [01:45.939](https://academa.ai/lectures/the-centres-of-a-triangle?t=105.93900000000001): radius\_formula (the "O C" part) is no longer emphasized.
- [01:45.939](https://academa.ai/lectures/the-centres-of-a-triangle?t=105.93900000000001): radius\_formula (the "R" part) is emphasized.
- [01:46.484](https://academa.ai/lectures/the-centres-of-a-triangle?t=106.4845): plane: retire a lit segment (unemphasize\_line).

##### [01:47.084](https://academa.ai/lectures/the-centres-of-a-triangle?t=107.08449999999999)

Narration: A circle with centre O and radius R therefore passes through A, B, and C. This is the circumcircle, and the gap from O back to G confirms that the circumcentre and centroid are different points.

Board: plane — a Figure (x\_range=(0.2, 7.9), y\_range=(0.0, 7.35), aspect=(7.7, 7.35)); triangle — a Polygon \[blue\] drawn in plane (vertices=((1.0, 2.0), (7.2, 2.2), (6.6, 6.0)), fill\_opacity=0.12); vertex\_a — a Point \[text\] labelled "A" drawn in plane (location=(1.0, 2.0)); vertex\_b — a Point \[text\] labelled "B" drawn in plane (location=(7.2, 2.2)); vertex\_c — a Point \[text\] labelled "C" drawn in plane (location=(6.6, 6.0)); centre\_o — a Point \[yellow\] labelled "O" drawn in plane (location=(4.050000000000001, 3.6499999999999995)); mid\_a — a Point \[text\] labelled "A'" drawn in plane (location=(6.9, 4.1)); mid\_b — a Point \[text\] labelled "B'" drawn in plane (location=(3.8, 4.0)); mid\_c — a Point \[text\] labelled "C'" drawn in plane (location=(4.1, 2.1)); perpendicular\_note — a Panel that says "A perpendicular bisector crosses a side at its midpoint and at a right angle. Every point on it is equally far from the side's ends."; radius\_formula — a Math \[text\] that says "$O A = O B = O C = R$"; heading\_o — a Heading that says "The Circumcentre"; bisector\_a — a Line \[gray\] drawn in plane (start=(7.196328889598721, 4.146788772041903), end=(3.640382214414073, 3.5853235075390635)); square\_a — an Angle \[gray\] drawn in plane (vertex=(6.9, 4.1), sides=((6.6, 6.0), (3.640382214414073, 3.5853235075390635)), right\_angle=True); bisector\_b — a Line \[gray\] drawn in plane (start=(3.625628541884271, 4.24412004136202), end=(5.718086039273018, 1.3146795450177744)); square\_b — an Angle \[gray\] drawn in plane (vertex=(3.8, 4.0), sides=((1.0, 2.0), (5.718086039273018, 1.3146795450177744)), right\_angle=True); bisector\_c — a Line \[gray\] drawn in plane (start=(4.109672388203287, 1.80015596569809), end=(3.993603729763838, 5.3982843773210085)); square\_c — an Angle \[gray\] drawn in plane (vertex=(4.1, 2.1), sides=((7.2, 2.2), (3.993603729763838, 5.3982843773210085)), right\_angle=True)

Actions:
- [01:47.084](https://academa.ai/lectures/the-centres-of-a-triangle?t=107.08449999999999): radius\_formula (the "R" part) is no longer emphasized.
- [01:47.433](https://academa.ai/lectures/the-centres-of-a-triangle?t=107.43299999999999): circumcircle is shown on the screen, drawn.
- [01:55.292](https://academa.ai/lectures/the-centres-of-a-triangle?t=115.292): The segment (4.050000000000001, 3.6499999999999995) to (4.933333333333333, 3.4) in plane is lit up.
- [01:56.453](https://academa.ai/lectures/the-centres-of-a-triangle?t=116.453): centre\_g is shown on the screen, written out.
- [01:57.487](https://academa.ai/lectures/the-centres-of-a-triangle?t=117.487): centre\_o is indicated — a transient flash.
- [01:58.393](https://academa.ai/lectures/the-centres-of-a-triangle?t=118.39299999999999): centre\_g is indicated — a transient flash.
- [02:0.378](https://academa.ai/lectures/the-centres-of-a-triangle?t=120.37799999999999): heading\_o is hidden from the screen — left the board.
- [02:0.378](https://academa.ai/lectures/the-centres-of-a-triangle?t=120.37799999999999): perpendicular\_note is hidden from the screen — left the board.
- [02:0.378](https://academa.ai/lectures/the-centres-of-a-triangle?t=120.37799999999999): radius\_formula is hidden from the screen — left the board.
- [02:0.378](https://academa.ai/lectures/the-centres-of-a-triangle?t=120.37799999999999): plane: retire a lit segment (unemphasize\_line).

##### [02:0.978](https://academa.ai/lectures/the-centres-of-a-triangle?t=120.978)

Narration: The third rule belongs to the corners. An angle bisector splits one corner into two equal angles.

Board: plane — a Figure (x\_range=(0.2, 7.9), y\_range=(0.0, 7.35), aspect=(7.7, 7.35)); triangle — a Polygon \[blue\] drawn in plane (vertices=((1.0, 2.0), (7.2, 2.2), (6.6, 6.0)), fill\_opacity=0.12); vertex\_a — a Point \[text\] labelled "A" drawn in plane (location=(1.0, 2.0)); vertex\_b — a Point \[text\] labelled "B" drawn in plane (location=(7.2, 2.2)); vertex\_c — a Point \[text\] labelled "C" drawn in plane (location=(6.6, 6.0)); centre\_g — a Point \[green\] labelled "G" drawn in plane (location=(4.933333333333333, 3.4)); centre\_o — a Point \[yellow\] labelled "O" drawn in plane (location=(4.050000000000001, 3.6499999999999995)); mid\_a — a Point \[text\] labelled "A'" drawn in plane (location=(6.9, 4.1)); mid\_b — a Point \[text\] labelled "B'" drawn in plane (location=(3.8, 4.0)); mid\_c — a Point \[text\] labelled "C'" drawn in plane (location=(4.1, 2.1)); bisector\_a — a Line \[gray\] drawn in plane (start=(7.196328889598721, 4.146788772041903), end=(3.640382214414073, 3.5853235075390635)); square\_a — an Angle \[gray\] drawn in plane (vertex=(6.9, 4.1), sides=((6.6, 6.0), (3.640382214414073, 3.5853235075390635)), right\_angle=True); bisector\_b — a Line \[gray\] drawn in plane (start=(3.625628541884271, 4.24412004136202), end=(5.718086039273018, 1.3146795450177744)); square\_b — an Angle \[gray\] drawn in plane (vertex=(3.8, 4.0), sides=((1.0, 2.0), (5.718086039273018, 1.3146795450177744)), right\_angle=True); bisector\_c — a Line \[gray\] drawn in plane (start=(4.109672388203287, 1.80015596569809), end=(3.993603729763838, 5.3982843773210085)); square\_c — an Angle \[gray\] drawn in plane (vertex=(4.1, 2.1), sides=((7.2, 2.2), (3.993603729763838, 5.3982843773210085)), right\_angle=True); circumcircle — a Circle \[yellow\] drawn in plane (center=(4.050000000000001, 3.650000000000001), radius=3.4677081768799414)

Actions:
- [02:0.978](https://academa.ai/lectures/the-centres-of-a-triangle?t=120.978): mid\_a is hidden from the screen.
- [02:0.978](https://academa.ai/lectures/the-centres-of-a-triangle?t=120.978): mid\_b is hidden from the screen.
- [02:0.978](https://academa.ai/lectures/the-centres-of-a-triangle?t=120.978): mid\_c is hidden from the screen.
- [02:0.978](https://academa.ai/lectures/the-centres-of-a-triangle?t=120.978): bisector\_a is hidden from the screen.
- [02:0.978](https://academa.ai/lectures/the-centres-of-a-triangle?t=120.978): bisector\_b is hidden from the screen.
- [02:0.978](https://academa.ai/lectures/the-centres-of-a-triangle?t=120.978): bisector\_c is hidden from the screen.
- [02:0.978](https://academa.ai/lectures/the-centres-of-a-triangle?t=120.978): square\_a is hidden from the screen.
- [02:0.978](https://academa.ai/lectures/the-centres-of-a-triangle?t=120.978): square\_b is hidden from the screen.
- [02:0.978](https://academa.ai/lectures/the-centres-of-a-triangle?t=120.978): square\_c is hidden from the screen.
- [02:0.978](https://academa.ai/lectures/the-centres-of-a-triangle?t=120.978): circumcircle is hidden from the screen.
- [02:0.978](https://academa.ai/lectures/the-centres-of-a-triangle?t=120.978): centre\_o is hidden from the screen.
- [02:0.978](https://academa.ai/lectures/the-centres-of-a-triangle?t=120.978): centre\_g is hidden from the screen.
- [02:0.978](https://academa.ai/lectures/the-centres-of-a-triangle?t=120.978): heading\_i is shown on the screen, written out.
- [02:1.814](https://academa.ai/lectures/the-centres-of-a-triangle?t=121.814): angle\_note is shown on the screen, written out.
- [02:4.275](https://academa.ai/lectures/the-centres-of-a-triangle?t=124.27499999999999): angle\_bisector\_b is shown on the screen, drawn.
- [02:4.275](https://academa.ai/lectures/the-centres-of-a-triangle?t=124.27499999999999): angle\_note (the "angle bisector" part) is emphasized.
- [02:6.063](https://academa.ai/lectures/the-centres-of-a-triangle?t=126.06299999999999): half\_b1 is shown on the screen, written out.
- [02:6.063](https://academa.ai/lectures/the-centres-of-a-triangle?t=126.06299999999999): half\_b2 is shown on the screen, written out.
- [02:6.319](https://academa.ai/lectures/the-centres-of-a-triangle?t=126.31899999999999): angle\_note (the "angle bisector" part) is no longer emphasized.
- [02:6.319](https://academa.ai/lectures/the-centres-of-a-triangle?t=126.31899999999999): angle\_note (the "equal angles" part) is emphasized.

##### [02:8.126](https://academa.ai/lectures/the-centres-of-a-triangle?t=128.126)

Narration: The angle bisectors from A and C meet the first at one point. This is the incentre I.

Board: plane — a Figure (x\_range=(0.2, 7.9), y\_range=(0.0, 7.35), aspect=(7.7, 7.35)); triangle — a Polygon \[blue\] drawn in plane (vertices=((1.0, 2.0), (7.2, 2.2), (6.6, 6.0)), fill\_opacity=0.12); vertex\_a — a Point \[text\] labelled "A" drawn in plane (location=(1.0, 2.0)); vertex\_b — a Point \[text\] labelled "B" drawn in plane (location=(7.2, 2.2)); vertex\_c — a Point \[text\] labelled "C" drawn in plane (location=(6.6, 6.0)); angle\_note — a Panel that says "An angle bisector splits a corner into two equal angles. Every point on it is equally far from the angle's two sides."; heading\_i — a Heading that says "The Incentre"; angle\_bisector\_b — a Line \[magenta\] drawn in plane (start=(7.2, 2.2), end=(4.456419576191015, 4.468871125850725)); half\_b1 — an Angle \[magenta\] drawn in plane (vertex=(7.2, 2.2), sides=((1.0, 2.0), (4.456419576191015, 4.468871125850725)), radius=0.65); half\_b2 — an Angle \[magenta\] drawn in plane (vertex=(7.2, 2.2), sides=((4.456419576191015, 4.468871125850725), (6.6, 6.0)), radius=0.65)

Actions:
- [02:8.126](https://academa.ai/lectures/the-centres-of-a-triangle?t=128.126): angle\_note (the "equal angles" part) is no longer emphasized.
- [02:9.438](https://academa.ai/lectures/the-centres-of-a-triangle?t=129.43800000000002): angle\_bisector\_a is shown on the screen, drawn.
- [02:10.077](https://academa.ai/lectures/the-centres-of-a-triangle?t=130.077): angle\_bisector\_c is shown on the screen, drawn.
- [02:11.4](https://academa.ai/lectures/the-centres-of-a-triangle?t=131.4): centre\_i is shown on the screen, written out.
- [02:13.641](https://academa.ai/lectures/the-centres-of-a-triangle?t=133.64100000000002): centre\_i is indicated — a transient flash.

##### [02:14.995](https://academa.ai/lectures/the-centres-of-a-triangle?t=134.9955)

Narration: The shortest distance from I to side A B is a perpendicular drop. The corresponding drops to B C and C A have exactly the same length. Their matching marks record the equality, and that common length is r.

Board: plane — a Figure (x\_range=(0.2, 7.9), y\_range=(0.0, 7.35), aspect=(7.7, 7.35)); triangle — a Polygon \[blue\] drawn in plane (vertices=((1.0, 2.0), (7.2, 2.2), (6.6, 6.0)), fill\_opacity=0.12); vertex\_a — a Point \[text\] labelled "A" drawn in plane (location=(1.0, 2.0)); vertex\_b — a Point \[text\] labelled "B" drawn in plane (location=(7.2, 2.2)); vertex\_c — a Point \[text\] labelled "C" drawn in plane (location=(6.6, 6.0)); centre\_i — a Point \[magenta\] labelled "I" drawn in plane (location=(5.57151267349114, 3.5467175380994163)); angle\_note — a Panel that says "An angle bisector splits a corner into two equal angles. Every point on it is equally far from the angle's two sides."; heading\_i — a Heading that says "The Incentre"; angle\_bisector\_b — a Line \[magenta\] drawn in plane (start=(7.2, 2.2), end=(4.456419576191015, 4.468871125850725)); half\_b1 — an Angle \[magenta\] drawn in plane (vertex=(7.2, 2.2), sides=((1.0, 2.0), (4.456419576191015, 4.468871125850725)), radius=0.65); half\_b2 — an Angle \[magenta\] drawn in plane (vertex=(7.2, 2.2), sides=((4.456419576191015, 4.468871125850725), (6.6, 6.0)), radius=0.65); angle\_bisector\_a — a Line \[magenta\] drawn in plane (start=(1.0, 2.0), end=(6.915558979628808, 4.001459795684212)); angle\_bisector\_c — a Line \[magenta\] drawn in plane (start=(6.6, 6.0), end=(4.97686492345502, 2.1282859652727426))

Actions:
- [02:17.178](https://academa.ai/lectures/the-centres-of-a-triangle?t=137.178): distance\_formula is shown on the screen, written out.
- [02:17.178](https://academa.ai/lectures/the-centres-of-a-triangle?t=137.178): drop\_c is shown on the screen, drawn.
- [02:17.178](https://academa.ai/lectures/the-centres-of-a-triangle?t=137.178): distance\_formula (the "d(I, A B)" part) is emphasized.
- [02:21.032](https://academa.ai/lectures/the-centres-of-a-triangle?t=141.03199999999998): drop\_a is shown on the screen, drawn.
- [02:21.032](https://academa.ai/lectures/the-centres-of-a-triangle?t=141.03199999999998): distance\_formula (the "d(I, A B)" part) is no longer emphasized.
- [02:21.032](https://academa.ai/lectures/the-centres-of-a-triangle?t=141.03199999999998): distance\_formula (the "d(I, B C)" part) is emphasized.
- [02:21.857](https://academa.ai/lectures/the-centres-of-a-triangle?t=141.85699999999997): drop\_b is shown on the screen, drawn.
- [02:21.857](https://academa.ai/lectures/the-centres-of-a-triangle?t=141.85699999999997): distance\_formula (the "d(I, B C)" part) is no longer emphasized.
- [02:21.857](https://academa.ai/lectures/the-centres-of-a-triangle?t=141.85699999999997): distance\_formula (the "d(I, C A)" part) is emphasized.
- [02:28.834](https://academa.ai/lectures/the-centres-of-a-triangle?t=148.834): distance\_formula (the "d(I, C A)" part) is no longer emphasized.
- [02:28.834](https://academa.ai/lectures/the-centres-of-a-triangle?t=148.834): distance\_formula (the "r" part) is emphasized.

##### [02:30.096](https://academa.ai/lectures/the-centres-of-a-triangle?t=150.0965)

Narration: The circle centred at I with radius r reaches every side and crosses none. It touches A B, B C, and C A at the three perpendicular feet. This is the incircle.

Board: plane — a Figure (x\_range=(0.2, 7.9), y\_range=(0.0, 7.35), aspect=(7.7, 7.35)); triangle — a Polygon \[blue\] drawn in plane (vertices=((1.0, 2.0), (7.2, 2.2), (6.6, 6.0)), fill\_opacity=0.12); vertex\_a — a Point \[text\] labelled "A" drawn in plane (location=(1.0, 2.0)); vertex\_b — a Point \[text\] labelled "B" drawn in plane (location=(7.2, 2.2)); vertex\_c — a Point \[text\] labelled "C" drawn in plane (location=(6.6, 6.0)); centre\_i — a Point \[magenta\] labelled "I" drawn in plane (location=(5.57151267349114, 3.5467175380994163)); angle\_note — a Panel that says "An angle bisector splits a corner into two equal angles. Every point on it is equally far from the angle's two sides."; distance\_formula — a Math \[text\] that says "$d(I, A B) = d(I, B C) = d(I, C A) = r$"; heading\_i — a Heading that says "The Incentre"; angle\_bisector\_b — a Line \[magenta\] drawn in plane (start=(7.2, 2.2), end=(4.456419576191015, 4.468871125850725)); half\_b1 — an Angle \[magenta\] drawn in plane (vertex=(7.2, 2.2), sides=((1.0, 2.0), (4.456419576191015, 4.468871125850725)), radius=0.65); half\_b2 — an Angle \[magenta\] drawn in plane (vertex=(7.2, 2.2), sides=((4.456419576191015, 4.468871125850725), (6.6, 6.0)), radius=0.65); angle\_bisector\_a — a Line \[magenta\] drawn in plane (start=(1.0, 2.0), end=(6.915558979628808, 4.001459795684212)); angle\_bisector\_c — a Line \[magenta\] drawn in plane (start=(6.6, 6.0), end=(4.97686492345502, 2.1282859652727426)); drop\_c — a Line \[yellow\] drawn in plane (start=(5.6166028304636875, 2.1489226719504417), end=(5.57151267349114, 3.5467175380994163), dashed=True); drop\_a — a Line \[yellow\] drawn in plane (start=(6.9529208497020365, 3.764834618553768), end=(5.57151267349114, 3.5467175380994163), dashed=True); drop\_b — a Line \[yellow\] drawn in plane (start=(4.758638308574938, 4.684741648982099), end=(5.57151267349114, 3.5467175380994163), dashed=True)

Actions:
- [02:30.096](https://academa.ai/lectures/the-centres-of-a-triangle?t=150.0965): distance\_formula (the "r" part) is no longer emphasized.
- [02:30.456](https://academa.ai/lectures/the-centres-of-a-triangle?t=150.45600000000002): incircle is shown on the screen, drawn.
- [02:35.901](https://academa.ai/lectures/the-centres-of-a-triangle?t=155.901): point\_3 is shown on the screen, grown.
- [02:36.447](https://academa.ai/lectures/the-centres-of-a-triangle?t=156.447): point\_4 is shown on the screen, grown.
- [02:37.178](https://academa.ai/lectures/the-centres-of-a-triangle?t=157.178): point\_5 is shown on the screen, grown.
- [02:37.901](https://academa.ai/lectures/the-centres-of-a-triangle?t=157.901): point\_3 is hidden from the screen.
- [02:38.447](https://academa.ai/lectures/the-centres-of-a-triangle?t=158.447): point\_4 is hidden from the screen.
- [02:39.178](https://academa.ai/lectures/the-centres-of-a-triangle?t=159.178): point\_5 is hidden from the screen.
- [02:41.102](https://academa.ai/lectures/the-centres-of-a-triangle?t=161.102): centre\_i is indicated — a transient flash.
- [02:42.019](https://academa.ai/lectures/the-centres-of-a-triangle?t=162.0195): angle\_note is hidden from the screen — left the board.
- [02:42.019](https://academa.ai/lectures/the-centres-of-a-triangle?t=162.0195): distance\_formula is hidden from the screen — left the board.
- [02:42.019](https://academa.ai/lectures/the-centres-of-a-triangle?t=162.0195): heading\_i is hidden from the screen — left the board.

##### [02:42.619](https://academa.ai/lectures/the-centres-of-a-triangle?t=162.6195)

Narration: The fourth rule returns to perpendicular lines. An altitude passes through a corner and meets the opposite side at a right angle.

Board: plane — a Figure (x\_range=(0.2, 7.9), y\_range=(0.0, 7.35), aspect=(7.7, 7.35)); triangle — a Polygon \[blue\] drawn in plane (vertices=((1.0, 2.0), (7.2, 2.2), (6.6, 6.0)), fill\_opacity=0.12); vertex\_a — a Point \[text\] labelled "A" drawn in plane (location=(1.0, 2.0)); vertex\_b — a Point \[text\] labelled "B" drawn in plane (location=(7.2, 2.2)); vertex\_c — a Point \[text\] labelled "C" drawn in plane (location=(6.6, 6.0)); centre\_i — a Point \[magenta\] labelled "I" drawn in plane (location=(5.57151267349114, 3.5467175380994163)); angle\_bisector\_b — a Line \[magenta\] drawn in plane (start=(7.2, 2.2), end=(4.456419576191015, 4.468871125850725)); half\_b1 — an Angle \[magenta\] drawn in plane (vertex=(7.2, 2.2), sides=((1.0, 2.0), (4.456419576191015, 4.468871125850725)), radius=0.65); half\_b2 — an Angle \[magenta\] drawn in plane (vertex=(7.2, 2.2), sides=((4.456419576191015, 4.468871125850725), (6.6, 6.0)), radius=0.65); angle\_bisector\_a — a Line \[magenta\] drawn in plane (start=(1.0, 2.0), end=(6.915558979628808, 4.001459795684212)); angle\_bisector\_c — a Line \[magenta\] drawn in plane (start=(6.6, 6.0), end=(4.97686492345502, 2.1282859652727426)); drop\_c — a Line \[yellow\] drawn in plane (start=(5.6166028304636875, 2.1489226719504417), end=(5.57151267349114, 3.5467175380994163), dashed=True); drop\_a — a Line \[yellow\] drawn in plane (start=(6.9529208497020365, 3.764834618553768), end=(5.57151267349114, 3.5467175380994163), dashed=True); drop\_b — a Line \[yellow\] drawn in plane (start=(4.758638308574938, 4.684741648982099), end=(5.57151267349114, 3.5467175380994163), dashed=True); incircle — a Circle \[magenta\] drawn in plane (center=(5.57151267349114, 3.5467175380994163), radius=1.398521937649975)

Actions:
- [02:42.619](https://academa.ai/lectures/the-centres-of-a-triangle?t=162.6195): angle\_bisector\_a is hidden from the screen.
- [02:42.619](https://academa.ai/lectures/the-centres-of-a-triangle?t=162.6195): angle\_bisector\_b is hidden from the screen.
- [02:42.619](https://academa.ai/lectures/the-centres-of-a-triangle?t=162.6195): angle\_bisector\_c is hidden from the screen.
- [02:42.619](https://academa.ai/lectures/the-centres-of-a-triangle?t=162.6195): half\_b1 is hidden from the screen.
- [02:42.619](https://academa.ai/lectures/the-centres-of-a-triangle?t=162.6195): half\_b2 is hidden from the screen.
- [02:42.619](https://academa.ai/lectures/the-centres-of-a-triangle?t=162.6195): centre\_i is hidden from the screen.
- [02:42.619](https://academa.ai/lectures/the-centres-of-a-triangle?t=162.6195): drop\_a is hidden from the screen.
- [02:42.619](https://academa.ai/lectures/the-centres-of-a-triangle?t=162.6195): drop\_b is hidden from the screen.
- [02:42.619](https://academa.ai/lectures/the-centres-of-a-triangle?t=162.6195): drop\_c is hidden from the screen.
- [02:42.619](https://academa.ai/lectures/the-centres-of-a-triangle?t=162.6195): incircle is hidden from the screen.
- [02:42.619](https://academa.ai/lectures/the-centres-of-a-triangle?t=162.6195): heading\_h is shown on the screen, written out.
- [02:43.339](https://academa.ai/lectures/the-centres-of-a-triangle?t=163.339): altitude\_note is shown on the screen, written out.
- [02:46.509](https://academa.ai/lectures/the-centres-of-a-triangle?t=166.509): altitude\_a is shown on the screen, drawn.
- [02:46.509](https://academa.ai/lectures/the-centres-of-a-triangle?t=166.509): altitude\_note (the "altitude" part) is emphasized.
- [02:49.968](https://academa.ai/lectures/the-centres-of-a-triangle?t=169.968): altitude\_square\_a is shown on the screen, written out.
- [02:49.968](https://academa.ai/lectures/the-centres-of-a-triangle?t=169.968): altitude\_note (the "altitude" part) is no longer emphasized.
- [02:49.968](https://academa.ai/lectures/the-centres-of-a-triangle?t=169.968): altitude\_note (the "right angle" part) is emphasized.

##### [02:51.636](https://academa.ai/lectures/the-centres-of-a-triangle?t=171.63649999999998)

Narration: The altitudes from B and C pass through the first altitude at one point. This fourth meeting point is the orthocentre H.

Board: plane — a Figure (x\_range=(0.2, 7.9), y\_range=(0.0, 7.35), aspect=(7.7, 7.35)); triangle — a Polygon \[blue\] drawn in plane (vertices=((1.0, 2.0), (7.2, 2.2), (6.6, 6.0)), fill\_opacity=0.12); vertex\_a — a Point \[text\] labelled "A" drawn in plane (location=(1.0, 2.0)); vertex\_b — a Point \[text\] labelled "B" drawn in plane (location=(7.2, 2.2)); vertex\_c — a Point \[text\] labelled "C" drawn in plane (location=(6.6, 6.0)); altitude\_note — a Panel that says "An altitude passes through a corner and meets the opposite side at a right angle. It is the height measured from that corner."; heading\_h — a Heading that says "The Orthocentre"; altitude\_a — a Line \[red\] drawn in plane (start=(7.08, 2.960000000000001), end=(1.0, 2.0)); altitude\_square\_a — an Angle \[red\] drawn in plane (vertex=(7.08, 2.960000000000001), sides=((6.6, 6.0), (1.0, 2.0)), right\_angle=True)

Actions:
- [02:51.636](https://academa.ai/lectures/the-centres-of-a-triangle?t=171.63649999999998): altitude\_note (the "right angle" part) is no longer emphasized.
- [02:52.739](https://academa.ai/lectures/the-centres-of-a-triangle?t=172.739): altitude\_b is shown on the screen, drawn.
- [02:52.739](https://academa.ai/lectures/the-centres-of-a-triangle?t=172.739): altitude\_square\_b is shown on the screen, written out.
- [02:53.285](https://academa.ai/lectures/the-centres-of-a-triangle?t=173.285): altitude\_c is shown on the screen, drawn.
- [02:53.285](https://academa.ai/lectures/the-centres-of-a-triangle?t=173.285): altitude\_square\_c is shown on the screen, written out.
- [02:55.363](https://academa.ai/lectures/the-centres-of-a-triangle?t=175.363): centre\_h is shown on the screen, written out.
- [02:58.718](https://academa.ai/lectures/the-centres-of-a-triangle?t=178.718): centre\_h is indicated — a transient flash.

##### [03:0.142](https://academa.ai/lectures/the-centres-of-a-triangle?t=180.14249999999998)

Narration: For this acute triangle, every altitude foot and the orthocentre lie inside. An obtuse angle changes that geometry.

Board: plane — a Figure (x\_range=(0.2, 7.9), y\_range=(0.0, 7.35), aspect=(7.7, 7.35)); triangle — a Polygon \[blue\] drawn in plane (vertices=((1.0, 2.0), (7.2, 2.2), (6.6, 6.0)), fill\_opacity=0.12); vertex\_a — a Point \[text\] labelled "A" drawn in plane (location=(1.0, 2.0)); vertex\_b — a Point \[text\] labelled "B" drawn in plane (location=(7.2, 2.2)); vertex\_c — a Point \[text\] labelled "C" drawn in plane (location=(6.6, 6.0)); centre\_h — a Point \[red\] labelled "H" drawn in plane (location=(6.699999999999999, 2.900000000000001)); altitude\_note — a Panel that says "An altitude passes through a corner and meets the opposite side at a right angle. It is the height measured from that corner."; heading\_h — a Heading that says "The Orthocentre"; altitude\_a — a Line \[red\] drawn in plane (start=(7.08, 2.960000000000001), end=(1.0, 2.0)); altitude\_square\_a — an Angle \[red\] drawn in plane (vertex=(7.08, 2.960000000000001), sides=((6.6, 6.0), (1.0, 2.0)), right\_angle=True); altitude\_b — a Line \[red\] drawn in plane (start=(5.2, 5.0), end=(7.2, 2.2)); altitude\_square\_b — an Angle \[red\] drawn in plane (vertex=(5.2, 5.0), sides=((1.0, 2.0), (7.2, 2.2)), right\_angle=True); altitude\_c — a Line \[red\] drawn in plane (start=(6.723076923076922, 2.184615384615385), end=(6.6, 6.0)); altitude\_square\_c — an Angle \[red\] drawn in plane (vertex=(6.723076923076922, 2.184615384615385), sides=((7.2, 2.2), (6.6, 6.0)), right\_angle=True)

Actions:
- [03:2.894](https://academa.ai/lectures/the-centres-of-a-triangle?t=182.89399999999998): point\_6 is shown on the screen, grown.
- [03:2.894](https://academa.ai/lectures/the-centres-of-a-triangle?t=182.89399999999998): point\_7 is shown on the screen, grown.
- [03:2.894](https://academa.ai/lectures/the-centres-of-a-triangle?t=182.89399999999998): point\_8 is shown on the screen, grown.
- [03:3.603](https://academa.ai/lectures/the-centres-of-a-triangle?t=183.60299999999998): centre\_h is indicated — a transient flash.
- [03:4.894](https://academa.ai/lectures/the-centres-of-a-triangle?t=184.89399999999998): point\_6 is hidden from the screen.
- [03:4.894](https://academa.ai/lectures/the-centres-of-a-triangle?t=184.89399999999998): point\_7 is hidden from the screen.
- [03:4.894](https://academa.ai/lectures/the-centres-of-a-triangle?t=184.89399999999998): point\_8 is hidden from the screen.
- [03:9.098](https://academa.ai/lectures/the-centres-of-a-triangle?t=189.09766666666664): altitude\_note is hidden from the screen — left the board.
- [03:9.098](https://academa.ai/lectures/the-centres-of-a-triangle?t=189.09766666666664): heading\_h is hidden from the screen — left the board.
- [03:9.098](https://academa.ai/lectures/the-centres-of-a-triangle?t=189.09766666666664): plane is hidden from the screen — left the board.
- [03:9.098](https://academa.ai/lectures/the-centres-of-a-triangle?t=189.09766666666664): triangle is hidden from the screen — plane left the board.
- [03:9.098](https://academa.ai/lectures/the-centres-of-a-triangle?t=189.09766666666664): vertex\_a is hidden from the screen — plane left the board.
- [03:9.098](https://academa.ai/lectures/the-centres-of-a-triangle?t=189.09766666666664): vertex\_b is hidden from the screen — plane left the board.
- [03:9.098](https://academa.ai/lectures/the-centres-of-a-triangle?t=189.09766666666664): vertex\_c is hidden from the screen — plane left the board.
- [03:9.098](https://academa.ai/lectures/the-centres-of-a-triangle?t=189.09766666666664): centre\_h is hidden from the screen — plane left the board.
- [03:9.098](https://academa.ai/lectures/the-centres-of-a-triangle?t=189.09766666666664): altitude\_a is hidden from the screen — plane left the board.
- [03:9.098](https://academa.ai/lectures/the-centres-of-a-triangle?t=189.09766666666664): altitude\_square\_a is hidden from the screen — plane left the board.
- [03:9.098](https://academa.ai/lectures/the-centres-of-a-triangle?t=189.09766666666664): altitude\_b is hidden from the screen — plane left the board.
- [03:9.098](https://academa.ai/lectures/the-centres-of-a-triangle?t=189.09766666666664): altitude\_square\_b is hidden from the screen — plane left the board.
- [03:9.098](https://academa.ai/lectures/the-centres-of-a-triangle?t=189.09766666666664): altitude\_c is hidden from the screen — plane left the board.
- [03:9.098](https://academa.ai/lectures/the-centres-of-a-triangle?t=189.09766666666664): altitude\_square\_c is hidden from the screen — plane left the board.

### Scene 2: [When the Orthocentre Moves Outside](https://academa.ai/lectures/the-centres-of-a-triangle?t=190.1393333333333)

Span: 03:10.139–04:2.731 (190.1393333333333s–242.73141666666663s).

#### Objects

- altitude\_p: a Line \[red\] drawn in plane (start=(0.8, 0.8), end=(6.208, 5.006222222222225))
- altitude\_q: a Line \[red\] drawn in plane (start=(7.4, 0.8), end=(5.944, 5.006222222222225))
- altitude\_r: a Line \[red\] drawn in plane (start=(6.0, 0.8), end=(6.0, 5.006222222222225))
- caption: a Tex \[text\] that says "Where do the three altitudes meet?"
- centre\_h: a Point \[red\] labelled "H" drawn in plane (location=(6.0, 4.844444444444447))
- extension\_qr: a Line \[gray\] drawn in plane (start=(6.0, 2.6), end=(4.74, 4.22), dashed=True)
- extension\_rp: a Line \[gray\] drawn in plane (start=(6.0, 2.6), end=(7.3, 3.05), dashed=True)
- heading: a Heading that says "When the Triangle is Obtuse"
- obtuse\_mark: an Angle \[yellow\] drawn in plane (vertex=(6.0, 2.6), sides=((7.4, 0.8), (0.8, 0.8)), radius=0.55)
- plane: a Figure (x\_range=(0.2, 7.9), y\_range=(0.3, 5.25), aspect=(7.7, 4.95))
- point: a Point \[yellow\] drawn in plane (location=(4.912307692307691, 3.998461538461539))
- point\_2: a Point \[yellow\] drawn in plane (location=(6.693791281373844, 2.8401585204755615))
- point\_3: a Point \[yellow\] drawn in plane (location=(6.0, 0.8))
- square\_p: an Angle \[red\] drawn in plane (vertex=(4.912307692307691, 3.998461538461539), sides=((6.0, 2.6), (0.8, 0.8)), right\_angle=True)
- square\_q: an Angle \[red\] drawn in plane (vertex=(6.693791281373844, 2.8401585204755615), sides=((6.0, 2.6), (7.4, 0.8)), right\_angle=True)
- square\_r: an Angle \[red\] drawn in plane (vertex=(6.0, 0.8), sides=((7.4, 0.8), (6.0, 2.6)), right\_angle=True)
- triangle: a Polygon \[blue\] drawn in plane (vertices=((0.8, 0.8), (7.4, 0.8), (6.0, 2.6)), fill\_opacity=0.12)
- vertex\_p: a Point \[text\] labelled "P" drawn in plane (location=(0.8, 0.8))
- vertex\_q: a Point \[text\] labelled "Q" drawn in plane (location=(7.4, 0.8))
- vertex\_r: a Point \[text\] labelled "R" drawn in plane (location=(6.0, 2.6))

#### Beats

##### [03:10.139](https://academa.ai/lectures/the-centres-of-a-triangle?t=190.1393333333333)

Narration: At R, the angle is greater than a right angle, so this triangle is obtuse. The question is still the same: where do its three altitudes meet?

Board: Empty.

Actions:
- [03:10.139](https://academa.ai/lectures/the-centres-of-a-triangle?t=190.1393333333333): heading is shown on the screen, written out.
- [03:10.139](https://academa.ai/lectures/the-centres-of-a-triangle?t=190.1393333333333): plane is shown on the screen, written out.
- [03:10.139](https://academa.ai/lectures/the-centres-of-a-triangle?t=190.1393333333333): triangle is shown on the screen, written out.
- [03:10.139](https://academa.ai/lectures/the-centres-of-a-triangle?t=190.1393333333333): vertex\_p is shown on the screen, written out.
- [03:10.139](https://academa.ai/lectures/the-centres-of-a-triangle?t=190.1393333333333): vertex\_q is shown on the screen, written out.
- [03:10.139](https://academa.ai/lectures/the-centres-of-a-triangle?t=190.1393333333333): vertex\_r is shown on the screen, written out.
- [03:11.312](https://academa.ai/lectures/the-centres-of-a-triangle?t=191.3123333333333): obtuse\_mark is shown on the screen, written out.
- [03:14.075](https://academa.ai/lectures/the-centres-of-a-triangle?t=194.0753333333333): obtuse\_mark is indicated — a transient flash.
- [03:15.433](https://academa.ai/lectures/the-centres-of-a-triangle?t=195.4333333333333): caption is shown on the screen, written out.

##### [03:20.027](https://academa.ai/lectures/the-centres-of-a-triangle?t=200.0273333333333)

Narration: Continue side Q R beyond R. The perpendicular from P meets that extended line outside the triangle.

Board: plane — a Figure (x\_range=(0.2, 7.9), y\_range=(0.3, 5.25), aspect=(7.7, 4.95)); caption — a Tex \[text\] that says "Where do the three altitudes meet?"; heading — a Heading that says "When the Triangle is Obtuse"; triangle — a Polygon \[blue\] drawn in plane (vertices=((0.8, 0.8), (7.4, 0.8), (6.0, 2.6)), fill\_opacity=0.12); vertex\_p — a Point \[text\] labelled "P" drawn in plane (location=(0.8, 0.8)); vertex\_q — a Point \[text\] labelled "Q" drawn in plane (location=(7.4, 0.8)); vertex\_r — a Point \[text\] labelled "R" drawn in plane (location=(6.0, 2.6)); obtuse\_mark — an Angle \[yellow\] drawn in plane (vertex=(6.0, 2.6), sides=((7.4, 0.8), (0.8, 0.8)), radius=0.55)

Actions:
- [03:20.503](https://academa.ai/lectures/the-centres-of-a-triangle?t=200.5033333333333): extension\_qr is shown on the screen, drawn.
- [03:23.51](https://academa.ai/lectures/the-centres-of-a-triangle?t=203.5103333333333): altitude\_p is shown on the screen, drawn.
- [03:25.925](https://academa.ai/lectures/the-centres-of-a-triangle?t=205.9253333333333): square\_p is shown on the screen, written out.
- [03:25.925](https://academa.ai/lectures/the-centres-of-a-triangle?t=205.9253333333333): point is shown on the screen, grown.
- [03:27.425](https://academa.ai/lectures/the-centres-of-a-triangle?t=207.4253333333333): point is hidden from the screen.

##### [03:28.069](https://academa.ai/lectures/the-centres-of-a-triangle?t=208.0693333333333)

Narration: Continue side R P beyond R as well. The perpendicular from Q has its own right-angle foot beyond the triangle.

Board: plane — a Figure (x\_range=(0.2, 7.9), y\_range=(0.3, 5.25), aspect=(7.7, 4.95)); caption — a Tex \[text\] that says "Where do the three altitudes meet?"; heading — a Heading that says "When the Triangle is Obtuse"; triangle — a Polygon \[blue\] drawn in plane (vertices=((0.8, 0.8), (7.4, 0.8), (6.0, 2.6)), fill\_opacity=0.12); vertex\_p — a Point \[text\] labelled "P" drawn in plane (location=(0.8, 0.8)); vertex\_q — a Point \[text\] labelled "Q" drawn in plane (location=(7.4, 0.8)); vertex\_r — a Point \[text\] labelled "R" drawn in plane (location=(6.0, 2.6)); obtuse\_mark — an Angle \[yellow\] drawn in plane (vertex=(6.0, 2.6), sides=((7.4, 0.8), (0.8, 0.8)), radius=0.55); extension\_qr — a Line \[gray\] drawn in plane (start=(6.0, 2.6), end=(4.74, 4.22), dashed=True); altitude\_p — a Line \[red\] drawn in plane (start=(0.8, 0.8), end=(6.208, 5.006222222222225)); square\_p — an Angle \[red\] drawn in plane (vertex=(4.912307692307691, 3.998461538461539), sides=((6.0, 2.6), (0.8, 0.8)), right\_angle=True)

Actions:
- [03:28.417](https://academa.ai/lectures/the-centres-of-a-triangle?t=208.4173333333333): extension\_rp is shown on the screen, drawn.
- [03:29.811](https://academa.ai/lectures/the-centres-of-a-triangle?t=209.8113333333333): point\_2 is shown on the screen, grown.
- [03:31.311](https://academa.ai/lectures/the-centres-of-a-triangle?t=211.3113333333333): point\_2 is hidden from the screen.
- [03:31.982](https://academa.ai/lectures/the-centres-of-a-triangle?t=211.9823333333333): altitude\_q is shown on the screen, drawn.
- [03:33.886](https://academa.ai/lectures/the-centres-of-a-triangle?t=213.88633333333328): square\_q is shown on the screen, written out.

##### [03:36.669](https://academa.ai/lectures/the-centres-of-a-triangle?t=216.66883333333328)

Narration: The altitude from R meets P Q on the side itself. Its right-angle mark is inside, but the line continues upward to meet the other two altitudes.

Board: plane — a Figure (x\_range=(0.2, 7.9), y\_range=(0.3, 5.25), aspect=(7.7, 4.95)); caption — a Tex \[text\] that says "Where do the three altitudes meet?"; heading — a Heading that says "When the Triangle is Obtuse"; triangle — a Polygon \[blue\] drawn in plane (vertices=((0.8, 0.8), (7.4, 0.8), (6.0, 2.6)), fill\_opacity=0.12); vertex\_p — a Point \[text\] labelled "P" drawn in plane (location=(0.8, 0.8)); vertex\_q — a Point \[text\] labelled "Q" drawn in plane (location=(7.4, 0.8)); vertex\_r — a Point \[text\] labelled "R" drawn in plane (location=(6.0, 2.6)); obtuse\_mark — an Angle \[yellow\] drawn in plane (vertex=(6.0, 2.6), sides=((7.4, 0.8), (0.8, 0.8)), radius=0.55); extension\_qr — a Line \[gray\] drawn in plane (start=(6.0, 2.6), end=(4.74, 4.22), dashed=True); altitude\_p — a Line \[red\] drawn in plane (start=(0.8, 0.8), end=(6.208, 5.006222222222225)); square\_p — an Angle \[red\] drawn in plane (vertex=(4.912307692307691, 3.998461538461539), sides=((6.0, 2.6), (0.8, 0.8)), right\_angle=True); extension\_rp — a Line \[gray\] drawn in plane (start=(6.0, 2.6), end=(7.3, 3.05), dashed=True); altitude\_q — a Line \[red\] drawn in plane (start=(7.4, 0.8), end=(5.944, 5.006222222222225)); square\_q — an Angle \[red\] drawn in plane (vertex=(6.693791281373844, 2.8401585204755615), sides=((6.0, 2.6), (7.4, 0.8)), right\_angle=True)

Actions:
- [03:37.214](https://academa.ai/lectures/the-centres-of-a-triangle?t=217.2143333333333): altitude\_r is shown on the screen, drawn.
- [03:41.115](https://academa.ai/lectures/the-centres-of-a-triangle?t=221.1153333333333): square\_r is shown on the screen, written out.
- [03:41.986](https://academa.ai/lectures/the-centres-of-a-triangle?t=221.98633333333328): point\_3 is shown on the screen, grown.
- [03:43.486](https://academa.ai/lectures/the-centres-of-a-triangle?t=223.48633333333328): point\_3 is hidden from the screen.

##### [03:46.789](https://academa.ai/lectures/the-centres-of-a-triangle?t=226.7893333333333)

Narration: All three altitudes still meet at one point. The orthocentre H lies outside an obtuse triangle.

Board: plane — a Figure (x\_range=(0.2, 7.9), y\_range=(0.3, 5.25), aspect=(7.7, 4.95)); caption — a Tex \[text\] that says "Where do the three altitudes meet?"; heading — a Heading that says "When the Triangle is Obtuse"; triangle — a Polygon \[blue\] drawn in plane (vertices=((0.8, 0.8), (7.4, 0.8), (6.0, 2.6)), fill\_opacity=0.12); vertex\_p — a Point \[text\] labelled "P" drawn in plane (location=(0.8, 0.8)); vertex\_q — a Point \[text\] labelled "Q" drawn in plane (location=(7.4, 0.8)); vertex\_r — a Point \[text\] labelled "R" drawn in plane (location=(6.0, 2.6)); obtuse\_mark — an Angle \[yellow\] drawn in plane (vertex=(6.0, 2.6), sides=((7.4, 0.8), (0.8, 0.8)), radius=0.55); extension\_qr — a Line \[gray\] drawn in plane (start=(6.0, 2.6), end=(4.74, 4.22), dashed=True); altitude\_p — a Line \[red\] drawn in plane (start=(0.8, 0.8), end=(6.208, 5.006222222222225)); square\_p — an Angle \[red\] drawn in plane (vertex=(4.912307692307691, 3.998461538461539), sides=((6.0, 2.6), (0.8, 0.8)), right\_angle=True); extension\_rp — a Line \[gray\] drawn in plane (start=(6.0, 2.6), end=(7.3, 3.05), dashed=True); altitude\_q — a Line \[red\] drawn in plane (start=(7.4, 0.8), end=(5.944, 5.006222222222225)); square\_q — an Angle \[red\] drawn in plane (vertex=(6.693791281373844, 2.8401585204755615), sides=((6.0, 2.6), (7.4, 0.8)), right\_angle=True); altitude\_r — a Line \[red\] drawn in plane (start=(6.0, 0.8), end=(6.0, 5.006222222222225)); square\_r — an Angle \[red\] drawn in plane (vertex=(6.0, 0.8), sides=((7.4, 0.8), (6.0, 2.6)), right\_angle=True)

Actions:
- [03:49.111](https://academa.ai/lectures/the-centres-of-a-triangle?t=229.11133333333328): centre\_h is shown on the screen, written out.
- [03:51.166](https://academa.ai/lectures/the-centres-of-a-triangle?t=231.16633333333328): centre\_h is indicated — a transient flash.
- [03:51.77](https://academa.ai/lectures/the-centres-of-a-triangle?t=231.7703333333333): caption becomes "The orthocentre lies outside the triangle.".

##### [03:54.46](https://academa.ai/lectures/the-centres-of-a-triangle?t=234.45983333333328)

Narration: The defining rule did not fail. Extending the two short sides exposed the same concurrency beyond the triangle.

Board: plane — a Figure (x\_range=(0.2, 7.9), y\_range=(0.3, 5.25), aspect=(7.7, 4.95)); caption — a Tex \[text\] that says "Where do the three altitudes meet?"; heading — a Heading that says "When the Triangle is Obtuse"; triangle — a Polygon \[blue\] drawn in plane (vertices=((0.8, 0.8), (7.4, 0.8), (6.0, 2.6)), fill\_opacity=0.12); vertex\_p — a Point \[text\] labelled "P" drawn in plane (location=(0.8, 0.8)); vertex\_q — a Point \[text\] labelled "Q" drawn in plane (location=(7.4, 0.8)); vertex\_r — a Point \[text\] labelled "R" drawn in plane (location=(6.0, 2.6)); obtuse\_mark — an Angle \[yellow\] drawn in plane (vertex=(6.0, 2.6), sides=((7.4, 0.8), (0.8, 0.8)), radius=0.55); extension\_qr — a Line \[gray\] drawn in plane (start=(6.0, 2.6), end=(4.74, 4.22), dashed=True); altitude\_p — a Line \[red\] drawn in plane (start=(0.8, 0.8), end=(6.208, 5.006222222222225)); square\_p — an Angle \[red\] drawn in plane (vertex=(4.912307692307691, 3.998461538461539), sides=((6.0, 2.6), (0.8, 0.8)), right\_angle=True); extension\_rp — a Line \[gray\] drawn in plane (start=(6.0, 2.6), end=(7.3, 3.05), dashed=True); altitude\_q — a Line \[red\] drawn in plane (start=(7.4, 0.8), end=(5.944, 5.006222222222225)); square\_q — an Angle \[red\] drawn in plane (vertex=(6.693791281373844, 2.8401585204755615), sides=((6.0, 2.6), (7.4, 0.8)), right\_angle=True); altitude\_r — a Line \[red\] drawn in plane (start=(6.0, 0.8), end=(6.0, 5.006222222222225)); square\_r — an Angle \[red\] drawn in plane (vertex=(6.0, 0.8), sides=((7.4, 0.8), (6.0, 2.6)), right\_angle=True); centre\_h — a Point \[red\] labelled "H" drawn in plane (location=(6.0, 4.844444444444447))

Actions:
- [03:59.568](https://academa.ai/lectures/the-centres-of-a-triangle?t=239.5683333333333): centre\_h is indicated — a transient flash.
- [04:1.69](https://academa.ai/lectures/the-centres-of-a-triangle?t=241.68974999999995): caption is hidden from the screen — left the board.
- [04:1.69](https://academa.ai/lectures/the-centres-of-a-triangle?t=241.68974999999995): heading is hidden from the screen — left the board.
- [04:1.69](https://academa.ai/lectures/the-centres-of-a-triangle?t=241.68974999999995): plane is hidden from the screen — left the board.
- [04:1.69](https://academa.ai/lectures/the-centres-of-a-triangle?t=241.68974999999995): triangle is hidden from the screen — plane left the board.
- [04:1.69](https://academa.ai/lectures/the-centres-of-a-triangle?t=241.68974999999995): vertex\_p is hidden from the screen — plane left the board.
- [04:1.69](https://academa.ai/lectures/the-centres-of-a-triangle?t=241.68974999999995): vertex\_q is hidden from the screen — plane left the board.
- [04:1.69](https://academa.ai/lectures/the-centres-of-a-triangle?t=241.68974999999995): vertex\_r is hidden from the screen — plane left the board.
- [04:1.69](https://academa.ai/lectures/the-centres-of-a-triangle?t=241.68974999999995): obtuse\_mark is hidden from the screen — plane left the board.
- [04:1.69](https://academa.ai/lectures/the-centres-of-a-triangle?t=241.68974999999995): extension\_qr is hidden from the screen — plane left the board.
- [04:1.69](https://academa.ai/lectures/the-centres-of-a-triangle?t=241.68974999999995): altitude\_p is hidden from the screen — plane left the board.
- [04:1.69](https://academa.ai/lectures/the-centres-of-a-triangle?t=241.68974999999995): square\_p is hidden from the screen — plane left the board.
- [04:1.69](https://academa.ai/lectures/the-centres-of-a-triangle?t=241.68974999999995): extension\_rp is hidden from the screen — plane left the board.
- [04:1.69](https://academa.ai/lectures/the-centres-of-a-triangle?t=241.68974999999995): altitude\_q is hidden from the screen — plane left the board.
- [04:1.69](https://academa.ai/lectures/the-centres-of-a-triangle?t=241.68974999999995): square\_q is hidden from the screen — plane left the board.
- [04:1.69](https://academa.ai/lectures/the-centres-of-a-triangle?t=241.68974999999995): altitude\_r is hidden from the screen — plane left the board.
- [04:1.69](https://academa.ai/lectures/the-centres-of-a-triangle?t=241.68974999999995): square\_r is hidden from the screen — plane left the board.
- [04:1.69](https://academa.ai/lectures/the-centres-of-a-triangle?t=241.68974999999995): centre\_h is hidden from the screen — plane left the board.

### Scene 3: [The Euler Line](https://academa.ai/lectures/the-centres-of-a-triangle?t=242.73141666666663)

Span: 04:2.731–05:38.525 (242.73141666666663s–338.52512499999995s).

#### Objects

- altitude\_a: a Line \[red\] drawn in plane (start=((7.2 + ((corner\_x - 7.2) \* ((((-6.2 \* (corner\_x - 7.2)) + (-0.…, end=(1.0, 2.0))
- altitude\_b: a Line \[red\] drawn in plane (start=((corner\_x + ((1.0 - corner\_x) \* (((((7.2 - corner\_x) \* (1.0 - …, end=(7.2, 2.2))
- altitude\_c: a Line \[red\] drawn in plane (start=((1.0 + (6.2 \* (((((corner\_x - 1.0) \* 6.2) + ((corner\_y - 2.0) …, end=(\<VariableNumber corner\_x = 6.6\>, \<VariableNumber corner\_y = 6.…)
- bisector\_a: a Line \[gray\] drawn in plane (start=((((7.2 + corner\_x) / 2.0) + (((-1.0 \* (corner\_y - 2.2)) / ((((…, end=(((((7.2 + corner\_x) / 2.0) + (((-1.0 \* (corner\_y - 2.2)) / (((…)
- bisector\_b: a Line \[gray\] drawn in plane (start=((((corner\_x + 1.0) / 2.0) + (((-1.0 \* (2.0 - corner\_y)) / ((((…, end=(((((corner\_x + 1.0) / 2.0) + (((-1.0 \* (2.0 - corner\_y)) / (((…)
- bisector\_c: a Line \[gray\] drawn in plane (start=(4.109672388203287, 1.80015596569809), end=(3.993603729763838, 5.3982843773210085))
- centre\_g: a Point \[green\] labelled "G" drawn in plane (location=(((8.2 + corner\_x) / 3.0), ((4.2 + corner\_y) / 3.0)))
- centre\_h: a Point \[red\] labelled "H" drawn in plane (location=(((7.2 + ((corner\_x - 7.2) \* ((((-6.2 \* (corner\_x - 7.2)) + (-0…)
- centre\_i: a Point \[magenta\] labelled "I" drawn in plane (location=(((((sqrt((((7.2 - corner\_x) \*\* 2.0) + ((2.2 - corner\_y) \*\* 2.0…)
- centre\_o: a Point \[yellow\] labelled "O" drawn in plane (location=(((((7.2 + corner\_x) / 2.0) + (((-1.0 \* (corner\_y - 2.2)) / (((…)
- circumcircle: a Circle \[yellow\] drawn in plane (center=(((((5.0 \* (2.2 - corner\_y)) + (56.68000000000001 \* (corner\_y -…, radius=((((1.0 - ((((5.0 \* (2.2 - corner\_y)) + (56.68000000000001 \* (c…)
- corner\_x: a VariableNumber (initial\_value=6.6)
- corner\_y: a VariableNumber (initial\_value=6.0)
- euler\_line: a Line \[text\] drawn in plane (start=((((((7.2 + corner\_x) / 2.0) + (((-1.0 \* (corner\_y - 2.2)) / ((…, end=((((((7.2 + corner\_x) / 2.0) + (((-1.0 \* (corner\_y - 2.2)) / ((…)
- euler\_note: a Panel that says "In every triangle, the circumcentre $O$, centroid $G$, and orthocentre $H$ are collinear. The centroid is one third of the way from $O$ to $H$."
- heading: a Heading that says "The Euler Line"
- median\_a: a Line \[green\] drawn in plane (start=(1.0, 2.0), end=(((7.2 + corner\_x) / 2.0), ((2.2 + corner\_y) / 2.0)))
- median\_b: a Line \[green\] drawn in plane (start=(7.2, 2.2), end=(((corner\_x + 1.0) / 2.0), ((corner\_y + 2.0) / 2.0)))
- median\_c: a Line \[green\] drawn in plane (start=(\<VariableNumber corner\_x = 6.6\>, \<VariableNumber corner\_y = 6.…, end=(4.1, 2.1))
- mid\_a: a Point \[text\] labelled "A'" drawn in plane (location=(((7.2 + corner\_x) / 2.0), ((2.2 + corner\_y) / 2.0)))
- mid\_b: a Point \[text\] labelled "B'" drawn in plane (location=(((corner\_x + 1.0) / 2.0), ((corner\_y + 2.0) / 2.0)))
- mid\_c: a Point \[text\] labelled "C'" drawn in plane (location=(4.1, 2.1))
- plane: a Figure (x\_range=(0.2, 7.9), y\_range=(-0.2, 7.7), aspect=(7.7, 7.9))
- ratio: a Math \[text\] that says "$O G : G H = 1 : 2$"
- triangle: a Polygon \[blue\] drawn in plane (vertices=((1.0, 2.0), (7.2, 2.2), (\<VariableNumber corner\_x = 6.6\>, \<Var…, fill\_opacity=0.12)
- vertex\_a: a Point \[text\] labelled "A" drawn in plane (location=(1.0, 2.0))
- vertex\_b: a Point \[text\] labelled "B" drawn in plane (location=(7.2, 2.2))
- vertex\_c: a Point \[text\] labelled "C" drawn in plane (location=(\<VariableNumber corner\_x = 6.6\>, \<VariableNumber corner\_y = 6.…)

#### Beats

##### [04:2.731](https://academa.ai/lectures/the-centres-of-a-triangle?t=242.73141666666663)

Narration: The four centres return on a triangle whose top corner can move. First, the three side midpoints mark where the medians land.

Board: Empty.

Actions:
- [04:2.731](https://academa.ai/lectures/the-centres-of-a-triangle?t=242.73141666666663): heading is shown on the screen, written out.
- [04:2.731](https://academa.ai/lectures/the-centres-of-a-triangle?t=242.73141666666663): plane is shown on the screen, written out.
- [04:2.731](https://academa.ai/lectures/the-centres-of-a-triangle?t=242.73141666666663): triangle is shown on the screen, written out.
- [04:2.731](https://academa.ai/lectures/the-centres-of-a-triangle?t=242.73141666666663): vertex\_a is shown on the screen, written out.
- [04:2.731](https://academa.ai/lectures/the-centres-of-a-triangle?t=242.73141666666663): vertex\_b is shown on the screen, written out.
- [04:2.731](https://academa.ai/lectures/the-centres-of-a-triangle?t=242.73141666666663): vertex\_c is shown on the screen, written out.
- [04:8.397](https://academa.ai/lectures/the-centres-of-a-triangle?t=248.39741666666663): mid\_a is shown on the screen, written out.
- [04:8.397](https://academa.ai/lectures/the-centres-of-a-triangle?t=248.39741666666663): mid\_b is shown on the screen, written out.
- [04:8.647](https://academa.ai/lectures/the-centres-of-a-triangle?t=248.64741666666663): mid\_c is shown on the screen, written out.

##### [04:11.481](https://academa.ai/lectures/the-centres-of-a-triangle?t=251.48141666666663)

Narration: The median from A reaches A prime, the median from B reaches B prime, and the median from C reaches C prime. Their meeting point is the centroid G.

Board: plane — a Figure (x\_range=(0.2, 7.9), y\_range=(-0.2, 7.7), aspect=(7.7, 7.9)); heading — a Heading that says "The Euler Line"; triangle — a Polygon \[blue\] drawn in plane (vertices=((1.0, 2.0), (7.2, 2.2), (\<VariableNumber corner\_x = 6.6\>, \<Var…, fill\_opacity=0.12); vertex\_a — a Point \[text\] labelled "A" drawn in plane (location=(1.0, 2.0)); vertex\_b — a Point \[text\] labelled "B" drawn in plane (location=(7.2, 2.2)); vertex\_c — a Point \[text\] labelled "C" drawn in plane (location=(\<VariableNumber corner\_x = 6.6\>, \<VariableNumber corner\_y = 6.…); mid\_a — a Point \[text\] labelled "A'" drawn in plane (location=(((7.2 + corner\_x) / 2.0), ((2.2 + corner\_y) / 2.0))); mid\_b — a Point \[text\] labelled "B'" drawn in plane (location=(((corner\_x + 1.0) / 2.0), ((corner\_y + 2.0) / 2.0))); mid\_c — a Point \[text\] labelled "C'" drawn in plane (location=(4.1, 2.1))

Actions:
- [04:12.538](https://academa.ai/lectures/the-centres-of-a-triangle?t=252.53841666666662): median\_a is shown on the screen, drawn.
- [04:14.906](https://academa.ai/lectures/the-centres-of-a-triangle?t=254.90641666666664): median\_b is shown on the screen, drawn.
- [04:17.298](https://academa.ai/lectures/the-centres-of-a-triangle?t=257.29841666666664): median\_c is shown on the screen, drawn.
- [04:20.096](https://academa.ai/lectures/the-centres-of-a-triangle?t=260.09641666666664): centre\_g is shown on the screen, written out.
- [04:21.129](https://academa.ai/lectures/the-centres-of-a-triangle?t=261.12941666666666): centre\_g is indicated — a transient flash.

##### [04:22.31](https://academa.ai/lectures/the-centres-of-a-triangle?t=262.30991666666665)

Narration: The perpendicular bisectors meet at the circumcentre O, the centre of the circle through all three corners.

Board: plane — a Figure (x\_range=(0.2, 7.9), y\_range=(-0.2, 7.7), aspect=(7.7, 7.9)); heading — a Heading that says "The Euler Line"; triangle — a Polygon \[blue\] drawn in plane (vertices=((1.0, 2.0), (7.2, 2.2), (\<VariableNumber corner\_x = 6.6\>, \<Var…, fill\_opacity=0.12); vertex\_a — a Point \[text\] labelled "A" drawn in plane (location=(1.0, 2.0)); vertex\_b — a Point \[text\] labelled "B" drawn in plane (location=(7.2, 2.2)); vertex\_c — a Point \[text\] labelled "C" drawn in plane (location=(\<VariableNumber corner\_x = 6.6\>, \<VariableNumber corner\_y = 6.…); mid\_a — a Point \[text\] labelled "A'" drawn in plane (location=(((7.2 + corner\_x) / 2.0), ((2.2 + corner\_y) / 2.0))); mid\_b — a Point \[text\] labelled "B'" drawn in plane (location=(((corner\_x + 1.0) / 2.0), ((corner\_y + 2.0) / 2.0))); mid\_c — a Point \[text\] labelled "C'" drawn in plane (location=(4.1, 2.1)); median\_a — a Line \[green\] drawn in plane (start=(1.0, 2.0), end=(((7.2 + corner\_x) / 2.0), ((2.2 + corner\_y) / 2.0))); median\_b — a Line \[green\] drawn in plane (start=(7.2, 2.2), end=(((corner\_x + 1.0) / 2.0), ((corner\_y + 2.0) / 2.0))); median\_c — a Line \[green\] drawn in plane (start=(\<VariableNumber corner\_x = 6.6\>, \<VariableNumber corner\_y = 6.…, end=(4.1, 2.1)); centre\_g — a Point \[green\] labelled "G" drawn in plane (location=(((8.2 + corner\_x) / 3.0), ((4.2 + corner\_y) / 3.0)))

Actions:
- [04:22.31](https://academa.ai/lectures/the-centres-of-a-triangle?t=262.30991666666665): median\_a is hidden from the screen.
- [04:22.31](https://academa.ai/lectures/the-centres-of-a-triangle?t=262.30991666666665): median\_b is hidden from the screen.
- [04:22.31](https://academa.ai/lectures/the-centres-of-a-triangle?t=262.30991666666665): median\_c is hidden from the screen.
- [04:22.31](https://academa.ai/lectures/the-centres-of-a-triangle?t=262.30991666666665): mid\_a is hidden from the screen.
- [04:22.31](https://academa.ai/lectures/the-centres-of-a-triangle?t=262.30991666666665): mid\_b is hidden from the screen.
- [04:22.31](https://academa.ai/lectures/the-centres-of-a-triangle?t=262.30991666666665): mid\_c is hidden from the screen.
- [04:23.285](https://academa.ai/lectures/the-centres-of-a-triangle?t=263.2854166666666): bisector\_a is shown on the screen, drawn.
- [04:23.285](https://academa.ai/lectures/the-centres-of-a-triangle?t=263.2854166666666): bisector\_b is shown on the screen, drawn.
- [04:23.535](https://academa.ai/lectures/the-centres-of-a-triangle?t=263.5354166666666): bisector\_c is shown on the screen, drawn.
- [04:25.154](https://academa.ai/lectures/the-centres-of-a-triangle?t=265.15441666666663): centre\_o is shown on the screen, written out.
- [04:25.154](https://academa.ai/lectures/the-centres-of-a-triangle?t=265.15441666666663): centre\_o is indicated — a transient flash.
- [04:26.605](https://academa.ai/lectures/the-centres-of-a-triangle?t=266.60541666666666): circumcircle is shown on the screen, drawn.

##### [04:29.341](https://academa.ai/lectures/the-centres-of-a-triangle?t=269.34141666666665)

Narration: The altitudes meet at the orthocentre H.

Board: plane — a Figure (x\_range=(0.2, 7.9), y\_range=(-0.2, 7.7), aspect=(7.7, 7.9)); heading — a Heading that says "The Euler Line"; triangle — a Polygon \[blue\] drawn in plane (vertices=((1.0, 2.0), (7.2, 2.2), (\<VariableNumber corner\_x = 6.6\>, \<Var…, fill\_opacity=0.12); vertex\_a — a Point \[text\] labelled "A" drawn in plane (location=(1.0, 2.0)); vertex\_b — a Point \[text\] labelled "B" drawn in plane (location=(7.2, 2.2)); vertex\_c — a Point \[text\] labelled "C" drawn in plane (location=(\<VariableNumber corner\_x = 6.6\>, \<VariableNumber corner\_y = 6.…); centre\_g — a Point \[green\] labelled "G" drawn in plane (location=(((8.2 + corner\_x) / 3.0), ((4.2 + corner\_y) / 3.0))); bisector\_a — a Line \[gray\] drawn in plane (start=((((7.2 + corner\_x) / 2.0) + (((-1.0 \* (corner\_y - 2.2)) / ((((…, end=(((((7.2 + corner\_x) / 2.0) + (((-1.0 \* (corner\_y - 2.2)) / (((…); bisector\_b — a Line \[gray\] drawn in plane (start=((((corner\_x + 1.0) / 2.0) + (((-1.0 \* (2.0 - corner\_y)) / ((((…, end=(((((corner\_x + 1.0) / 2.0) + (((-1.0 \* (2.0 - corner\_y)) / (((…); bisector\_c — a Line \[gray\] drawn in plane (start=(4.109672388203287, 1.80015596569809), end=(3.993603729763838, 5.3982843773210085)); centre\_o — a Point \[yellow\] labelled "O" drawn in plane (location=(((((7.2 + corner\_x) / 2.0) + (((-1.0 \* (corner\_y - 2.2)) / (((…); circumcircle — a Circle \[yellow\] drawn in plane (center=(((((5.0 \* (2.2 - corner\_y)) + (56.68000000000001 \* (corner\_y -…, radius=((((1.0 - ((((5.0 \* (2.2 - corner\_y)) + (56.68000000000001 \* (c…)

Actions:
- [04:29.341](https://academa.ai/lectures/the-centres-of-a-triangle?t=269.34141666666665): bisector\_a is hidden from the screen.
- [04:29.341](https://academa.ai/lectures/the-centres-of-a-triangle?t=269.34141666666665): bisector\_b is hidden from the screen.
- [04:29.341](https://academa.ai/lectures/the-centres-of-a-triangle?t=269.34141666666665): bisector\_c is hidden from the screen.
- [04:29.341](https://academa.ai/lectures/the-centres-of-a-triangle?t=269.34141666666665): circumcircle is hidden from the screen.
- [04:29.852](https://academa.ai/lectures/the-centres-of-a-triangle?t=269.8524166666666): altitude\_a is shown on the screen, drawn.
- [04:29.852](https://academa.ai/lectures/the-centres-of-a-triangle?t=269.8524166666666): altitude\_b is shown on the screen, drawn.
- [04:30.102](https://academa.ai/lectures/the-centres-of-a-triangle?t=270.1024166666666): altitude\_c is shown on the screen, drawn.
- [04:31.501](https://academa.ai/lectures/the-centres-of-a-triangle?t=271.5014166666666): centre\_h is shown on the screen, written out.
- [04:31.501](https://academa.ai/lectures/the-centres-of-a-triangle?t=271.5014166666666): centre\_h is indicated — a transient flash.

##### [04:32.867](https://academa.ai/lectures/the-centres-of-a-triangle?t=272.86741666666666)

Narration: The construction lines can now leave.

Board: plane — a Figure (x\_range=(0.2, 7.9), y\_range=(-0.2, 7.7), aspect=(7.7, 7.9)); heading — a Heading that says "The Euler Line"; triangle — a Polygon \[blue\] drawn in plane (vertices=((1.0, 2.0), (7.2, 2.2), (\<VariableNumber corner\_x = 6.6\>, \<Var…, fill\_opacity=0.12); vertex\_a — a Point \[text\] labelled "A" drawn in plane (location=(1.0, 2.0)); vertex\_b — a Point \[text\] labelled "B" drawn in plane (location=(7.2, 2.2)); vertex\_c — a Point \[text\] labelled "C" drawn in plane (location=(\<VariableNumber corner\_x = 6.6\>, \<VariableNumber corner\_y = 6.…); centre\_g — a Point \[green\] labelled "G" drawn in plane (location=(((8.2 + corner\_x) / 3.0), ((4.2 + corner\_y) / 3.0))); centre\_o — a Point \[yellow\] labelled "O" drawn in plane (location=(((((7.2 + corner\_x) / 2.0) + (((-1.0 \* (corner\_y - 2.2)) / (((…); altitude\_a — a Line \[red\] drawn in plane (start=((7.2 + ((corner\_x - 7.2) \* ((((-6.2 \* (corner\_x - 7.2)) + (-0.…, end=(1.0, 2.0)); altitude\_b — a Line \[red\] drawn in plane (start=((corner\_x + ((1.0 - corner\_x) \* (((((7.2 - corner\_x) \* (1.0 - …, end=(7.2, 2.2)); altitude\_c — a Line \[red\] drawn in plane (start=((1.0 + (6.2 \* (((((corner\_x - 1.0) \* 6.2) + ((corner\_y - 2.0) …, end=(\<VariableNumber corner\_x = 6.6\>, \<VariableNumber corner\_y = 6.…); centre\_h — a Point \[red\] labelled "H" drawn in plane (location=(((7.2 + ((corner\_x - 7.2) \* ((((-6.2 \* (corner\_x - 7.2)) + (-0…)

Actions:
- [04:32.867](https://academa.ai/lectures/the-centres-of-a-triangle?t=272.86741666666666): altitude\_a is hidden from the screen.
- [04:33.067](https://academa.ai/lectures/the-centres-of-a-triangle?t=273.06741666666665): altitude\_b is hidden from the screen.
- [04:33.267](https://academa.ai/lectures/the-centres-of-a-triangle?t=273.26741666666663): altitude\_c is hidden from the screen.

##### [04:35.963](https://academa.ai/lectures/the-centres-of-a-triangle?t=275.9634166666666)

Narration: Only three labelled points remain: the circumcentre O, the centroid G, and the orthocentre H.

Board: plane — a Figure (x\_range=(0.2, 7.9), y\_range=(-0.2, 7.7), aspect=(7.7, 7.9)); heading — a Heading that says "The Euler Line"; triangle — a Polygon \[blue\] drawn in plane (vertices=((1.0, 2.0), (7.2, 2.2), (\<VariableNumber corner\_x = 6.6\>, \<Var…, fill\_opacity=0.12); vertex\_a — a Point \[text\] labelled "A" drawn in plane (location=(1.0, 2.0)); vertex\_b — a Point \[text\] labelled "B" drawn in plane (location=(7.2, 2.2)); vertex\_c — a Point \[text\] labelled "C" drawn in plane (location=(\<VariableNumber corner\_x = 6.6\>, \<VariableNumber corner\_y = 6.…); centre\_g — a Point \[green\] labelled "G" drawn in plane (location=(((8.2 + corner\_x) / 3.0), ((4.2 + corner\_y) / 3.0))); centre\_o — a Point \[yellow\] labelled "O" drawn in plane (location=(((((7.2 + corner\_x) / 2.0) + (((-1.0 \* (corner\_y - 2.2)) / (((…); centre\_h — a Point \[red\] labelled "H" drawn in plane (location=(((7.2 + ((corner\_x - 7.2) \* ((((-6.2 \* (corner\_x - 7.2)) + (-0…)

Actions:
- [04:39.226](https://academa.ai/lectures/the-centres-of-a-triangle?t=279.22641666666664): centre\_o is indicated — a transient flash.
- [04:40.41](https://academa.ai/lectures/the-centres-of-a-triangle?t=280.4104166666666): centre\_g is indicated — a transient flash.
- [04:42.059](https://academa.ai/lectures/the-centres-of-a-triangle?t=282.0594166666666): centre\_h is indicated — a transient flash.

##### [04:43.28](https://academa.ai/lectures/the-centres-of-a-triangle?t=283.2799166666666)

Narration: They lie on one straight line, exactly. This is the Euler line.

Board: Unchanged from the preceding beat in this scene.

Actions:
- [04:43.28](https://academa.ai/lectures/the-centres-of-a-triangle?t=283.2799166666666): plane moves to a new place on the board.
- [04:43.28](https://academa.ai/lectures/the-centres-of-a-triangle?t=283.2799166666666): euler\_note is shown on the screen, written out.
- [04:44.551](https://academa.ai/lectures/the-centres-of-a-triangle?t=284.5514166666666): euler\_note (the "collinear." part) is emphasized.
- [04:44.946](https://academa.ai/lectures/the-centres-of-a-triangle?t=284.9464166666666): euler\_line is shown on the screen, drawn.

##### [04:48.982](https://academa.ai/lectures/the-centres-of-a-triangle?t=288.9824166666666)

Narration: That alignment belongs to the triangle, not to this one shape. As the top corner moves left, O, G, and H move with it and remain collinear.

Board: euler\_note — a Panel that says "In every triangle, the circumcentre $O$, centroid $G$, and orthocentre $H$ are collinear. The centroid is one third of the way from $O$ to $H$."; plane — a Figure (x\_range=(0.2, 7.9), y\_range=(-0.2, 7.7), aspect=(7.7, 7.9)); heading — a Heading that says "The Euler Line"; triangle — a Polygon \[blue\] drawn in plane (vertices=((1.0, 2.0), (7.2, 2.2), (\<VariableNumber corner\_x = 6.6\>, \<Var…, fill\_opacity=0.12); vertex\_a — a Point \[text\] labelled "A" drawn in plane (location=(1.0, 2.0)); vertex\_b — a Point \[text\] labelled "B" drawn in plane (location=(7.2, 2.2)); vertex\_c — a Point \[text\] labelled "C" drawn in plane (location=(\<VariableNumber corner\_x = 6.6\>, \<VariableNumber corner\_y = 6.…); centre\_g — a Point \[green\] labelled "G" drawn in plane (location=(((8.2 + corner\_x) / 3.0), ((4.2 + corner\_y) / 3.0))); centre\_o — a Point \[yellow\] labelled "O" drawn in plane (location=(((((7.2 + corner\_x) / 2.0) + (((-1.0 \* (corner\_y - 2.2)) / (((…); centre\_h — a Point \[red\] labelled "H" drawn in plane (location=(((7.2 + ((corner\_x - 7.2) \* ((((-6.2 \* (corner\_x - 7.2)) + (-0…); euler\_line — a Line \[text\] drawn in plane (start=((((((7.2 + corner\_x) / 2.0) + (((-1.0 \* (corner\_y - 2.2)) / ((…, end=((((((7.2 + corner\_x) / 2.0) + (((-1.0 \* (corner\_y - 2.2)) / ((…)

Actions:
- [04:48.982](https://academa.ai/lectures/the-centres-of-a-triangle?t=288.9824166666666): euler\_note (the "collinear." part) is no longer emphasized.
- [04:54.776](https://academa.ai/lectures/the-centres-of-a-triangle?t=294.77641666666665): triangle is redrawn as the numbers it depends on change.
- [04:54.776](https://academa.ai/lectures/the-centres-of-a-triangle?t=294.77641666666665): vertex\_c is redrawn as the numbers it depends on change.
- [04:54.776](https://academa.ai/lectures/the-centres-of-a-triangle?t=294.77641666666665): centre\_g is redrawn as the numbers it depends on change.
- [04:54.776](https://academa.ai/lectures/the-centres-of-a-triangle?t=294.77641666666665): centre\_o is redrawn as the numbers it depends on change.
- [04:54.776](https://academa.ai/lectures/the-centres-of-a-triangle?t=294.77641666666665): centre\_h is redrawn as the numbers it depends on change.
- [04:54.776](https://academa.ai/lectures/the-centres-of-a-triangle?t=294.77641666666665): euler\_line is redrawn as the numbers it depends on change.
- [04:54.776](https://academa.ai/lectures/the-centres-of-a-triangle?t=294.77641666666665): centre\_i is redrawn as the numbers it depends on change.
- [04:54.776](https://academa.ai/lectures/the-centres-of-a-triangle?t=294.77641666666665): corner\_x ticks to 5.3.
- [04:54.776](https://academa.ai/lectures/the-centres-of-a-triangle?t=294.77641666666665): corner\_y ticks to 6.5.
- [04:55.67](https://academa.ai/lectures/the-centres-of-a-triangle?t=295.67041666666665): centre\_o is indicated — a transient flash.
- [04:56.007](https://academa.ai/lectures/the-centres-of-a-triangle?t=296.00741666666664): centre\_g is indicated — a transient flash.
- [04:56.483](https://academa.ai/lectures/the-centres-of-a-triangle?t=296.48341666666664): centre\_h is indicated — a transient flash.

##### [04:59.608](https://academa.ai/lectures/the-centres-of-a-triangle?t=299.6079166666666)

Narration: Move the corner back to the right, and the same line carries all three centres again.

Board: Unchanged from the preceding beat in this scene.

Actions:
- [05:1.088](https://academa.ai/lectures/the-centres-of-a-triangle?t=301.0884166666666): triangle is redrawn as the numbers it depends on change.
- [05:1.088](https://academa.ai/lectures/the-centres-of-a-triangle?t=301.0884166666666): vertex\_c is redrawn as the numbers it depends on change.
- [05:1.088](https://academa.ai/lectures/the-centres-of-a-triangle?t=301.0884166666666): centre\_g is redrawn as the numbers it depends on change.
- [05:1.088](https://academa.ai/lectures/the-centres-of-a-triangle?t=301.0884166666666): centre\_o is redrawn as the numbers it depends on change.
- [05:1.088](https://academa.ai/lectures/the-centres-of-a-triangle?t=301.0884166666666): centre\_h is redrawn as the numbers it depends on change.
- [05:1.088](https://academa.ai/lectures/the-centres-of-a-triangle?t=301.0884166666666): euler\_line is redrawn as the numbers it depends on change.
- [05:1.088](https://academa.ai/lectures/the-centres-of-a-triangle?t=301.0884166666666): centre\_i is redrawn as the numbers it depends on change.
- [05:1.088](https://academa.ai/lectures/the-centres-of-a-triangle?t=301.0884166666666): corner\_x ticks to 6.6.
- [05:1.088](https://academa.ai/lectures/the-centres-of-a-triangle?t=301.0884166666666): corner\_y ticks to 6.0.
- [05:2.516](https://academa.ai/lectures/the-centres-of-a-triangle?t=302.5164166666666): euler\_line is indicated — a transient flash.

##### [05:5.537](https://academa.ai/lectures/the-centres-of-a-triangle?t=305.5369166666666)

Narration: Their spacing is fixed as well. O G is one part, while G H is two parts, so O G to G H is one to two.

Board: Unchanged from the preceding beat in this scene.

Actions:
- [05:8.329](https://academa.ai/lectures/the-centres-of-a-triangle?t=308.3294166666666): ratio is shown on the screen, written out.
- [05:8.329](https://academa.ai/lectures/the-centres-of-a-triangle?t=308.3294166666666): ratio (the "O G" part) is emphasized.
- [05:8.329](https://academa.ai/lectures/the-centres-of-a-triangle?t=308.3294166666666): The segment (4.050000000000001, 3.6499999999999995) to (4.933333333333333, 3.4) in plane is lit up.
- [05:9.153](https://academa.ai/lectures/the-centres-of-a-triangle?t=309.15341666666666): ratio (the "1" part) is emphasized.
- [05:9.153](https://academa.ai/lectures/the-centres-of-a-triangle?t=309.15341666666666): ratio (the "O G" part) is no longer emphasized.
- [05:10.291](https://academa.ai/lectures/the-centres-of-a-triangle?t=310.29141666666663): ratio (the "1" part) is no longer emphasized.
- [05:10.291](https://academa.ai/lectures/the-centres-of-a-triangle?t=310.29141666666663): ratio (the "G H" part) is emphasized.
- [05:10.291](https://academa.ai/lectures/the-centres-of-a-triangle?t=310.29141666666663): plane: retire a lit segment (unemphasize\_line).
- [05:10.291](https://academa.ai/lectures/the-centres-of-a-triangle?t=310.29141666666663): The segment (4.933333333333333, 3.4) to (6.699999999999999, 2.900000000000001) in plane is lit up.
- [05:10.965](https://academa.ai/lectures/the-centres-of-a-triangle?t=310.9654166666666): ratio (the "2" part) is emphasized.
- [05:10.965](https://academa.ai/lectures/the-centres-of-a-triangle?t=310.9654166666666): ratio (the "G H" part) is no longer emphasized.
- [05:14.807](https://academa.ai/lectures/the-centres-of-a-triangle?t=314.80741666666665): plane: retire a lit segment (unemphasize\_line).

##### [05:15.407](https://academa.ai/lectures/the-centres-of-a-triangle?t=315.4074166666666)

Narration: The incentre I also moves with the triangle, but it does not generally lie on the Euler line.

Board: euler\_note — a Panel that says "In every triangle, the circumcentre $O$, centroid $G$, and orthocentre $H$ are collinear. The centroid is one third of the way from $O$ to $H$."; ratio — a Math \[text\] that says "$O G : G H = 1 : 2$"; plane — a Figure (x\_range=(0.2, 7.9), y\_range=(-0.2, 7.7), aspect=(7.7, 7.9)); heading — a Heading that says "The Euler Line"; triangle — a Polygon \[blue\] drawn in plane (vertices=((1.0, 2.0), (7.2, 2.2), (\<VariableNumber corner\_x = 6.6\>, \<Var…, fill\_opacity=0.12); vertex\_a — a Point \[text\] labelled "A" drawn in plane (location=(1.0, 2.0)); vertex\_b — a Point \[text\] labelled "B" drawn in plane (location=(7.2, 2.2)); vertex\_c — a Point \[text\] labelled "C" drawn in plane (location=(\<VariableNumber corner\_x = 6.6\>, \<VariableNumber corner\_y = 6.…); centre\_g — a Point \[green\] labelled "G" drawn in plane (location=(((8.2 + corner\_x) / 3.0), ((4.2 + corner\_y) / 3.0))); centre\_o — a Point \[yellow\] labelled "O" drawn in plane (location=(((((7.2 + corner\_x) / 2.0) + (((-1.0 \* (corner\_y - 2.2)) / (((…); centre\_h — a Point \[red\] labelled "H" drawn in plane (location=(((7.2 + ((corner\_x - 7.2) \* ((((-6.2 \* (corner\_x - 7.2)) + (-0…); euler\_line — a Line \[text\] drawn in plane (start=((((((7.2 + corner\_x) / 2.0) + (((-1.0 \* (corner\_y - 2.2)) / ((…, end=((((((7.2 + corner\_x) / 2.0) + (((-1.0 \* (corner\_y - 2.2)) / ((…)

Actions:
- [05:15.407](https://academa.ai/lectures/the-centres-of-a-triangle?t=315.4074166666666): ratio (the "2" part) is no longer emphasized.
- [05:16.452](https://academa.ai/lectures/the-centres-of-a-triangle?t=316.45241666666664): centre\_i is shown on the screen, written out.
- [05:16.452](https://academa.ai/lectures/the-centres-of-a-triangle?t=316.45241666666664): centre\_i is indicated — a transient flash.
- [05:18.82](https://academa.ai/lectures/the-centres-of-a-triangle?t=318.82041666666663): centre\_i is indicated — a transient flash.

##### [05:21.743](https://academa.ai/lectures/the-centres-of-a-triangle?t=321.7429166666666)

Narration: The centroid G comes from medians, the circumcentre O from perpendicular bisectors, the incentre I from angle bisectors, and the orthocentre H from altitudes. Four independent rules reveal a remarkably organised triangle.

Board: euler\_note — a Panel that says "In every triangle, the circumcentre $O$, centroid $G$, and orthocentre $H$ are collinear. The centroid is one third of the way from $O$ to $H$."; ratio — a Math \[text\] that says "$O G : G H = 1 : 2$"; plane — a Figure (x\_range=(0.2, 7.9), y\_range=(-0.2, 7.7), aspect=(7.7, 7.9)); heading — a Heading that says "The Euler Line"; triangle — a Polygon \[blue\] drawn in plane (vertices=((1.0, 2.0), (7.2, 2.2), (\<VariableNumber corner\_x = 6.6\>, \<Var…, fill\_opacity=0.12); vertex\_a — a Point \[text\] labelled "A" drawn in plane (location=(1.0, 2.0)); vertex\_b — a Point \[text\] labelled "B" drawn in plane (location=(7.2, 2.2)); vertex\_c — a Point \[text\] labelled "C" drawn in plane (location=(\<VariableNumber corner\_x = 6.6\>, \<VariableNumber corner\_y = 6.…); centre\_g — a Point \[green\] labelled "G" drawn in plane (location=(((8.2 + corner\_x) / 3.0), ((4.2 + corner\_y) / 3.0))); centre\_o — a Point \[yellow\] labelled "O" drawn in plane (location=(((((7.2 + corner\_x) / 2.0) + (((-1.0 \* (corner\_y - 2.2)) / (((…); centre\_h — a Point \[red\] labelled "H" drawn in plane (location=(((7.2 + ((corner\_x - 7.2) \* ((((-6.2 \* (corner\_x - 7.2)) + (-0…); euler\_line — a Line \[text\] drawn in plane (start=((((((7.2 + corner\_x) / 2.0) + (((-1.0 \* (corner\_y - 2.2)) / ((…, end=((((((7.2 + corner\_x) / 2.0) + (((-1.0 \* (corner\_y - 2.2)) / ((…); centre\_i — a Point \[magenta\] labelled "I" drawn in plane (location=(((((sqrt((((7.2 - corner\_x) \*\* 2.0) + ((2.2 - corner\_y) \*\* 2.0…)

Actions:
- [05:22.764](https://academa.ai/lectures/the-centres-of-a-triangle?t=322.76441666666665): centre\_g is indicated — a transient flash.
- [05:25.342](https://academa.ai/lectures/the-centres-of-a-triangle?t=325.3424166666666): centre\_o is indicated — a transient flash.
- [05:28.395](https://academa.ai/lectures/the-centres-of-a-triangle?t=328.3954166666666): centre\_i is indicated — a transient flash.
- [05:31.193](https://academa.ai/lectures/the-centres-of-a-triangle?t=331.1934166666666): centre\_h is indicated — a transient flash.
- [05:35.883](https://academa.ai/lectures/the-centres-of-a-triangle?t=335.8834166666666): euler\_line is indicated — a transient flash.
- [05:37.483](https://academa.ai/lectures/the-centres-of-a-triangle?t=337.48345833333326): euler\_note is hidden from the screen — left the board.
- [05:37.483](https://academa.ai/lectures/the-centres-of-a-triangle?t=337.48345833333326): heading is hidden from the screen — left the board.
- [05:37.483](https://academa.ai/lectures/the-centres-of-a-triangle?t=337.48345833333326): plane is hidden from the screen — left the board.
- [05:37.483](https://academa.ai/lectures/the-centres-of-a-triangle?t=337.48345833333326): triangle is hidden from the screen — plane left the board.
- [05:37.483](https://academa.ai/lectures/the-centres-of-a-triangle?t=337.48345833333326): vertex\_a is hidden from the screen — plane left the board.
- [05:37.483](https://academa.ai/lectures/the-centres-of-a-triangle?t=337.48345833333326): vertex\_b is hidden from the screen — plane left the board.
- [05:37.483](https://academa.ai/lectures/the-centres-of-a-triangle?t=337.48345833333326): vertex\_c is hidden from the screen — plane left the board.
- [05:37.483](https://academa.ai/lectures/the-centres-of-a-triangle?t=337.48345833333326): centre\_g is hidden from the screen — plane left the board.
- [05:37.483](https://academa.ai/lectures/the-centres-of-a-triangle?t=337.48345833333326): centre\_o is hidden from the screen — plane left the board.
- [05:37.483](https://academa.ai/lectures/the-centres-of-a-triangle?t=337.48345833333326): centre\_h is hidden from the screen — plane left the board.
- [05:37.483](https://academa.ai/lectures/the-centres-of-a-triangle?t=337.48345833333326): euler\_line is hidden from the screen — plane left the board.
- [05:37.483](https://academa.ai/lectures/the-centres-of-a-triangle?t=337.48345833333326): centre\_i is hidden from the screen — plane left the board.
- [05:37.483](https://academa.ai/lectures/the-centres-of-a-triangle?t=337.48345833333326): ratio is hidden from the screen — left the board.