{"version":1,"lectureId":"01M14TZJ8RS7SN9K6VECDBV527","attempt":1,"publication":{"slug":"the-centres-of-a-triangle","title":"The Centres of a Triangle","subject":"mathematics","summary":"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.","metaDescription":"Medians, perpendicular bisectors, angle bisectors and altitudes: four rules, four centres, and the straight line three of them share.","transcript":"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. 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. 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.","watch":{"version":1,"scenes":[{"title":"Four Constructions","start":0,"end":190.1393333333333,"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. 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Four different geometric rules locate four special points, and the surprise is that each rule makes three independently defined lines meet.","live":[],"does":[[0,"card is shown on the screen, written out."],[1.5,"card: enter:write-left-to-right."],[11.494,"card is hidden from the screen — left the board."]]},{"start":12.693999999999999,"say":"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.","live":null,"does":[[12.693999999999999,"heading_centres is shown on the screen, written out."],[12.693999999999999,"plane is shown on the screen, written out."],[12.693999999999999,"triangle is shown on the screen, written out."],[12.693999999999999,"vertex_a is shown on the screen, written out."],[12.693999999999999,"vertex_b is shown on the screen, written out."],[12.693999999999999,"vertex_c is shown on the screen, written out."],[13.517999999999999,"destination_note is shown on the screen, written out."],[14.957999999999998,"centre_g is shown on the screen, written out."],[14.957999999999998,"centre_g is indicated — a transient flash."],[16.572,"centre_o is shown on the screen, written out."],[16.572,"centre_o is indicated — a transient flash."],[18.035,"centre_i is shown on the screen, written out."],[18.035,"centre_i is indicated — a transient flash."],[19.765,"centre_h is shown on the screen, written out."],[19.765,"centre_h is indicated — a transient flash."]]},{"start":26.4135,"say":"The first rule begins at the midpoint of every side. Those three landing places are A prime, B prime, and C prime.","live":["destination_note","plane","heading_centres","triangle","vertex_a","vertex_b","vertex_c","centre_g","centre_o","centre_i","centre_h"],"does":[[26.4135,"centre_g is hidden from the screen."],[26.4135,"centre_o is hidden from the screen."],[26.4135,"centre_i is hidden from the screen."],[26.4135,"centre_h is hidden from the screen."],[31.545,"mid_a is shown on the screen, written out."],[32.288000000000004,"mid_b is shown on the screen, written out."],[33.438,"mid_c is shown on the screen, written out."],[34.5055,"destination_note is hidden from the screen — left the board."],[34.5055,"heading_centres is hidden from the screen — left the board."]]},{"start":35.1055,"say":"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.","live":["plane","triangle","vertex_a","vertex_b","vertex_c","mid_a","mid_b","mid_c"],"does":[[35.1055,"heading_g is shown on the screen, written out."],[35.1055,"median_note is shown on the screen, written out."],[36.034,"median_a is shown on the screen, drawn."],[38.055,"median_note (the \"median\" part) is emphasized."],[42.315,"median_b is shown on the screen, drawn."]]},{"start":44.831,"say":"Those two meet. The third median, from C to C prime, passes through the very same point. That common point is the centroid G.","live":["plane","triangle","vertex_a","vertex_b","vertex_c","mid_a","mid_b","mid_c","median_note","heading_g","median_a","median_b"],"does":[[44.831,"median_note (the \"median\" part) is no longer emphasized."],[46.932,"median_c is shown on the screen, drawn."],[50.729,"centre_g is shown on the screen, written out."],[53.341,"centre_g is indicated — a transient flash."]]},{"start":55.0445,"say":"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.","live":["plane","triangle","vertex_a","vertex_b","vertex_c","centre_g","mid_a","mid_b","mid_c","median_note","heading_g","median_a","median_b","median_c"],"does":[[56.669000000000004,"point is shown on the screen, grown."],[58.169000000000004,"point is hidden from the screen."],[60.466,"ratio_g is shown on the screen, written out."],[60.466,"ratio_g (the \"A G\" part) is emphasized."],[60.466,"The segment (1.0, 2.0) to (4.933333333333333, 3.4) in plane is lit up."],[61.197,"ratio_g (the \"2\" part) is emphasized."],[61.197,"ratio_g (the \"A G\" part) is no longer emphasized."],[63.148,"ratio_g (the \"2\" part) is no longer emphasized."],[63.148,"ratio_g (the \"G A'\" part) is emphasized."],[63.148,"plane: retire a lit segment (unemphasize_line)."],[63.148,"The segment (4.933333333333333, 3.4) to (6.9, 4.1) in plane is lit up."],[64.239,"ratio_g (the \"1\" part) is emphasized."],[64.239,"ratio_g (the \"G A'\" part) is no longer emphasized."],[68.8365,"heading_g is hidden from the screen — left the board."],[68.8365,"median_note is hidden from the screen — left the board."],[68.8365,"ratio_g is hidden from the screen — left the board."],[68.8365,"ratio_g (the \"1\" part) is no longer emphasized."],[68.8365,"plane: retire a lit segment (unemphasize_line)."]]},{"start":69.4365,"say":"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.","live":["plane","triangle","vertex_a","vertex_b","vertex_c","centre_g","mid_a","mid_b","mid_c","median_a","median_b","median_c"],"does":[[69.4365,"median_a is hidden from the screen."],[69.4365,"median_b is hidden from the screen."],[69.4365,"median_c is hidden from the screen."],[69.4365,"centre_g is hidden from the screen."],[69.4365,"heading_o is shown on the screen, written out."],[70.31899999999999,"perpendicular_note is shown on the screen, written out."],[70.598,"bisector_a is shown on the screen, drawn."],[73.13999999999999,"square_a is shown on the screen, written out."],[73.13999999999999,"perpendicular_note (the \"right angle\" part) is emphasized."],[75.544,"perpendicular_note (the \"perpendicular bisector\" part) is emphasized."],[75.544,"perpendicular_note (the \"right angle\" part) is no longer emphasized."]]},{"start":80.497,"say":"Choose any point on that line. Its distance to B equals its distance to C, the two ends of the side it bisects.","live":["plane","triangle","vertex_a","vertex_b","vertex_c","mid_a","mid_b","mid_c","perpendicular_note","heading_o","bisector_a","square_a"],"does":[[80.497,"perpendicular_note (the \"perpendicular bisector\" part) is no longer emphasized."],[81.449,"point_2 is shown on the screen, grown."],[83.449,"point_2 is hidden from the screen."],[83.98,"The segment (4.991641950984239, 3.798680308050143) to (7.2, 2.2) in plane is lit up."],[85.42,"The segment (4.991641950984239, 3.798680308050143) to (6.6, 6.0) in plane is lit up."],[85.42,"plane: retire a lit segment (unemphasize_line)."],[88.392,"plane: retire a lit segment (unemphasize_line)."]]},{"start":88.992,"say":"The other two perpendicular bisectors carry the same promise for their sides. All three pass through one point, the circumcentre O.","live":null,"does":[[89.50300000000001,"bisector_b is shown on the screen, drawn."],[89.50300000000001,"square_b is shown on the screen, written out."],[89.712,"bisector_c is shown on the screen, drawn."],[89.712,"square_c is shown on the screen, written out."],[95.9,"centre_o is shown on the screen, written out."],[96.608,"centre_o is indicated — a transient flash."]]},{"start":97.7825,"say":"Because O lies on all three bisectors, O A, O B, and O C are equal. Call their common length R.","live":["plane","triangle","vertex_a","vertex_b","vertex_c","centre_o","mid_a","mid_b","mid_c","perpendicular_note","heading_o","bisector_a","square_a","bisector_b","square_b","bisector_c","square_c"],"does":[[100.85300000000001,"radius_formula is shown on the screen, written out."],[100.85300000000001,"radius_formula (the \"O A\" part) is emphasized."],[100.85300000000001,"The segment (4.050000000000001, 3.6499999999999995) to (1.0, 2.0) in plane is lit up."],[101.68900000000001,"radius_formula (the \"O A\" part) is no longer emphasized."],[101.68900000000001,"radius_formula (the \"O B\" part) is emphasized."],[101.68900000000001,"plane: retire a lit segment (unemphasize_line)."],[101.68900000000001,"The segment (4.050000000000001, 3.6499999999999995) to (7.2, 2.2) in plane is lit up."],[102.641,"radius_formula (the \"O B\" part) is no longer emphasized."],[102.641,"radius_formula (the \"O C\" part) is emphasized."],[102.641,"The segment (4.050000000000001, 3.6499999999999995) to (6.6, 6.0) in plane is lit up."],[102.641,"plane: retire a lit segment (unemphasize_line)."],[105.93900000000001,"radius_formula (the \"O C\" part) is no longer emphasized."],[105.93900000000001,"radius_formula (the \"R\" part) is emphasized."],[106.4845,"plane: retire a lit segment (unemphasize_line)."]]},{"start":107.08449999999999,"say":"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.","live":["plane","triangle","vertex_a","vertex_b","vertex_c","centre_o","mid_a","mid_b","mid_c","perpendicular_note","radius_formula","heading_o","bisector_a","square_a","bisector_b","square_b","bisector_c","square_c"],"does":[[107.08449999999999,"radius_formula (the \"R\" part) is no longer emphasized."],[107.43299999999999,"circumcircle is shown on the screen, drawn."],[115.292,"The segment (4.050000000000001, 3.6499999999999995) to (4.933333333333333, 3.4) in plane is lit up."],[116.453,"centre_g is shown on the screen, written out."],[117.487,"centre_o is indicated — a transient flash."],[118.39299999999999,"centre_g is indicated — a transient flash."],[120.37799999999999,"heading_o is hidden from the screen — left the board."],[120.37799999999999,"perpendicular_note is hidden from the screen — left the board."],[120.37799999999999,"radius_formula is hidden from the screen — left the board."],[120.37799999999999,"plane: retire a lit segment (unemphasize_line)."]]},{"start":120.978,"say":"The third rule belongs to the corners. An angle bisector splits one corner into two equal angles.","live":["plane","triangle","vertex_a","vertex_b","vertex_c","centre_g","centre_o","mid_a","mid_b","mid_c","bisector_a","square_a","bisector_b","square_b","bisector_c","square_c","circumcircle"],"does":[[120.978,"mid_a is hidden from the screen."],[120.978,"mid_b is hidden from the screen."],[120.978,"mid_c is hidden from the screen."],[120.978,"bisector_a is hidden from the screen."],[120.978,"bisector_b is hidden from the screen."],[120.978,"bisector_c is hidden from the screen."],[120.978,"square_a is hidden from the screen."],[120.978,"square_b is hidden from the screen."],[120.978,"square_c is hidden from the screen."],[120.978,"circumcircle is hidden from the screen."],[120.978,"centre_o is hidden from the screen."],[120.978,"centre_g is hidden from the screen."],[120.978,"heading_i is shown on the screen, written out."],[121.814,"angle_note is shown on the screen, written out."],[124.27499999999999,"angle_bisector_b is shown on the screen, drawn."],[124.27499999999999,"angle_note (the \"angle bisector\" part) is emphasized."],[126.06299999999999,"half_b1 is shown on the screen, written out."],[126.06299999999999,"half_b2 is shown on the screen, written out."],[126.31899999999999,"angle_note (the \"angle bisector\" part) is no longer emphasized."],[126.31899999999999,"angle_note (the \"equal angles\" part) is emphasized."]]},{"start":128.126,"say":"The angle bisectors from A and C meet the first at one point. This is the incentre I.","live":["plane","triangle","vertex_a","vertex_b","vertex_c","angle_note","heading_i","angle_bisector_b","half_b1","half_b2"],"does":[[128.126,"angle_note (the \"equal angles\" part) is no longer emphasized."],[129.43800000000002,"angle_bisector_a is shown on the screen, drawn."],[130.077,"angle_bisector_c is shown on the screen, drawn."],[131.4,"centre_i is shown on the screen, written out."],[133.64100000000002,"centre_i is indicated — a transient flash."]]},{"start":134.9955,"say":"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.","live":["plane","triangle","vertex_a","vertex_b","vertex_c","centre_i","angle_note","heading_i","angle_bisector_b","half_b1","half_b2","angle_bisector_a","angle_bisector_c"],"does":[[137.178,"distance_formula is shown on the screen, written out."],[137.178,"drop_c is shown on the screen, drawn."],[137.178,"distance_formula (the \"d(I, A B)\" part) is emphasized."],[141.03199999999998,"drop_a is shown on the screen, drawn."],[141.03199999999998,"distance_formula (the \"d(I, A B)\" part) is no longer emphasized."],[141.03199999999998,"distance_formula (the \"d(I, B C)\" part) is emphasized."],[141.85699999999997,"drop_b is shown on the screen, drawn."],[141.85699999999997,"distance_formula (the \"d(I, B C)\" part) is no longer emphasized."],[141.85699999999997,"distance_formula (the \"d(I, C A)\" part) is emphasized."],[148.834,"distance_formula (the \"d(I, C A)\" part) is no longer emphasized."],[148.834,"distance_formula (the \"r\" part) is emphasized."]]},{"start":150.0965,"say":"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.","live":["plane","triangle","vertex_a","vertex_b","vertex_c","centre_i","angle_note","distance_formula","heading_i","angle_bisector_b","half_b1","half_b2","angle_bisector_a","angle_bisector_c","drop_c","drop_a","drop_b"],"does":[[150.0965,"distance_formula (the \"r\" part) is no longer emphasized."],[150.45600000000002,"incircle is shown on the screen, drawn."],[155.901,"point_3 is shown on the screen, grown."],[156.447,"point_4 is shown on the screen, grown."],[157.178,"point_5 is shown on the screen, grown."],[157.901,"point_3 is hidden from the screen."],[158.447,"point_4 is hidden from the screen."],[159.178,"point_5 is hidden from the screen."],[161.102,"centre_i is indicated — a transient flash."],[162.0195,"angle_note is hidden from the screen — left the board."],[162.0195,"distance_formula is hidden from the screen — left the board."],[162.0195,"heading_i is hidden from the screen — left the board."]]},{"start":162.6195,"say":"The fourth rule returns to perpendicular lines. An altitude passes through a corner and meets the opposite side at a right angle.","live":["plane","triangle","vertex_a","vertex_b","vertex_c","centre_i","angle_bisector_b","half_b1","half_b2","angle_bisector_a","angle_bisector_c","drop_c","drop_a","drop_b","incircle"],"does":[[162.6195,"angle_bisector_a is hidden from the screen."],[162.6195,"angle_bisector_b is hidden from the screen."],[162.6195,"angle_bisector_c is hidden from the screen."],[162.6195,"half_b1 is hidden from the screen."],[162.6195,"half_b2 is hidden from the screen."],[162.6195,"centre_i is hidden from the screen."],[162.6195,"drop_a is hidden from the screen."],[162.6195,"drop_b is hidden from the screen."],[162.6195,"drop_c is hidden from the screen."],[162.6195,"incircle is hidden from the screen."],[162.6195,"heading_h is shown on the screen, written out."],[163.339,"altitude_note is shown on the screen, written out."],[166.509,"altitude_a is shown on the screen, drawn."],[166.509,"altitude_note (the \"altitude\" part) is emphasized."],[169.968,"altitude_square_a is shown on the screen, written out."],[169.968,"altitude_note (the \"altitude\" part) is no longer emphasized."],[169.968,"altitude_note (the \"right angle\" part) is emphasized."]]},{"start":171.63649999999998,"say":"The altitudes from B and C pass through the first altitude at one point. This fourth meeting point is the orthocentre H.","live":["plane","triangle","vertex_a","vertex_b","vertex_c","altitude_note","heading_h","altitude_a","altitude_square_a"],"does":[[171.63649999999998,"altitude_note (the \"right angle\" part) is no longer emphasized."],[172.739,"altitude_b is shown on the screen, drawn."],[172.739,"altitude_square_b is shown on the screen, written out."],[173.285,"altitude_c is shown on the screen, drawn."],[173.285,"altitude_square_c is shown on the screen, written out."],[175.363,"centre_h is shown on the screen, written out."],[178.718,"centre_h is indicated — a transient flash."]]},{"start":180.14249999999998,"say":"For this acute triangle, every altitude foot and the orthocentre lie inside. An obtuse angle changes that geometry.","live":["plane","triangle","vertex_a","vertex_b","vertex_c","centre_h","altitude_note","heading_h","altitude_a","altitude_square_a","altitude_b","altitude_square_b","altitude_c","altitude_square_c"],"does":[[182.89399999999998,"point_6 is shown on the screen, grown."],[182.89399999999998,"point_7 is shown on the screen, grown."],[182.89399999999998,"point_8 is shown on the screen, grown."],[183.60299999999998,"centre_h is indicated — a transient flash."],[184.89399999999998,"point_6 is hidden from the screen."],[184.89399999999998,"point_7 is hidden from the screen."],[184.89399999999998,"point_8 is hidden from the screen."],[189.09766666666664,"altitude_note is hidden from the screen — left the board."],[189.09766666666664,"heading_h is hidden from the screen — left the board."],[189.09766666666664,"plane is hidden from the screen — left the board."],[189.09766666666664,"triangle is hidden from the screen — plane left the board."],[189.09766666666664,"vertex_a is hidden from the screen — plane left the board."],[189.09766666666664,"vertex_b is hidden from the screen — plane left the board."],[189.09766666666664,"vertex_c is hidden from the screen — plane left the board."],[189.09766666666664,"centre_h is hidden from the screen — plane left the board."],[189.09766666666664,"altitude_a is hidden from the screen — plane left the board."],[189.09766666666664,"altitude_square_a is hidden from the screen — plane left the board."],[189.09766666666664,"altitude_b is hidden from the screen — plane left the board."],[189.09766666666664,"altitude_square_b is hidden from the screen — plane left the board."],[189.09766666666664,"altitude_c is hidden from the screen — plane left the board."],[189.09766666666664,"altitude_square_c is hidden from the screen — plane left the board."]]}]},{"title":"When the Orthocentre Moves Outside","start":190.1393333333333,"end":242.73141666666663,"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":[{"start":190.1393333333333,"say":"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?","live":[],"does":[[190.1393333333333,"heading is shown on the screen, written out."],[190.1393333333333,"plane is shown on the screen, written out."],[190.1393333333333,"triangle is shown on the screen, written out."],[190.1393333333333,"vertex_p is shown on the screen, written out."],[190.1393333333333,"vertex_q is shown on the screen, written out."],[190.1393333333333,"vertex_r is shown on the screen, written out."],[191.3123333333333,"obtuse_mark is shown on the screen, written out."],[194.0753333333333,"obtuse_mark is indicated — a transient flash."],[195.4333333333333,"caption is shown on the screen, written out."]]},{"start":200.0273333333333,"say":"Continue side Q R beyond R. The perpendicular from P meets that extended line outside the triangle.","live":["plane","caption","heading","triangle","vertex_p","vertex_q","vertex_r","obtuse_mark"],"does":[[200.5033333333333,"extension_qr is shown on the screen, drawn."],[203.5103333333333,"altitude_p is shown on the screen, drawn."],[205.9253333333333,"square_p is shown on the screen, written out."],[205.9253333333333,"point is shown on the screen, grown."],[207.4253333333333,"point is hidden from the screen."]]},{"start":208.0693333333333,"say":"Continue side R P beyond R as well. The perpendicular from Q has its own right-angle foot beyond the triangle.","live":["plane","caption","heading","triangle","vertex_p","vertex_q","vertex_r","obtuse_mark","extension_qr","altitude_p","square_p"],"does":[[208.4173333333333,"extension_rp is shown on the screen, drawn."],[209.8113333333333,"point_2 is shown on the screen, grown."],[211.3113333333333,"point_2 is hidden from the screen."],[211.9823333333333,"altitude_q is shown on the screen, drawn."],[213.88633333333328,"square_q is shown on the screen, written out."]]},{"start":216.66883333333328,"say":"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.","live":["plane","caption","heading","triangle","vertex_p","vertex_q","vertex_r","obtuse_mark","extension_qr","altitude_p","square_p","extension_rp","altitude_q","square_q"],"does":[[217.2143333333333,"altitude_r is shown on the screen, drawn."],[221.1153333333333,"square_r is shown on the screen, written out."],[221.98633333333328,"point_3 is shown on the screen, grown."],[223.48633333333328,"point_3 is hidden from the screen."]]},{"start":226.7893333333333,"say":"All three altitudes still meet at one point. The orthocentre H lies outside an obtuse triangle.","live":["plane","caption","heading","triangle","vertex_p","vertex_q","vertex_r","obtuse_mark","extension_qr","altitude_p","square_p","extension_rp","altitude_q","square_q","altitude_r","square_r"],"does":[[229.11133333333328,"centre_h is shown on the screen, written out."],[231.16633333333328,"centre_h is indicated — a transient flash."],[231.7703333333333,"caption becomes \"The orthocentre lies outside the triangle.\"."]]},{"start":234.45983333333328,"say":"The defining rule did not fail. Extending the two short sides exposed the same concurrency beyond the triangle.","live":["plane","caption","heading","triangle","vertex_p","vertex_q","vertex_r","obtuse_mark","extension_qr","altitude_p","square_p","extension_rp","altitude_q","square_q","altitude_r","square_r","centre_h"],"does":[[239.5683333333333,"centre_h is indicated — a transient flash."],[241.68974999999995,"caption is hidden from the screen — left the board."],[241.68974999999995,"heading is hidden from the screen — left the board."],[241.68974999999995,"plane is hidden from the screen — left the board."],[241.68974999999995,"triangle is hidden from the screen — plane left the board."],[241.68974999999995,"vertex_p is hidden from the screen — plane left the board."],[241.68974999999995,"vertex_q is hidden from the screen — plane left the board."],[241.68974999999995,"vertex_r is hidden from the screen — plane left the board."],[241.68974999999995,"obtuse_mark is hidden from the screen — plane left the board."],[241.68974999999995,"extension_qr is hidden from the screen — plane left the board."],[241.68974999999995,"altitude_p is hidden from the screen — plane left the board."],[241.68974999999995,"square_p is hidden from the screen — plane left the board."],[241.68974999999995,"extension_rp is hidden from the screen — plane left the board."],[241.68974999999995,"altitude_q is hidden from the screen — plane left the board."],[241.68974999999995,"square_q is hidden from the screen — plane left the board."],[241.68974999999995,"altitude_r is hidden from the screen — plane left the board."],[241.68974999999995,"square_r is hidden from the screen — plane left the board."],[241.68974999999995,"centre_h is hidden from the screen — plane left the board."]]}]},{"title":"The Euler Line","start":242.73141666666663,"end":338.52512499999995,"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. 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First, the three side midpoints mark where the medians land.","live":[],"does":[[242.73141666666663,"heading is shown on the screen, written out."],[242.73141666666663,"plane is shown on the screen, written out."],[242.73141666666663,"triangle is shown on the screen, written out."],[242.73141666666663,"vertex_a is shown on the screen, written out."],[242.73141666666663,"vertex_b is shown on the screen, written out."],[242.73141666666663,"vertex_c is shown on the screen, written out."],[248.39741666666663,"mid_a is shown on the screen, written out."],[248.39741666666663,"mid_b is shown on the screen, written out."],[248.64741666666663,"mid_c is shown on the screen, written out."]]},{"start":251.48141666666663,"say":"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.","live":["plane","heading","triangle","vertex_a","vertex_b","vertex_c","mid_a","mid_b","mid_c"],"does":[[252.53841666666662,"median_a is shown on the screen, drawn."],[254.90641666666664,"median_b is shown on the screen, drawn."],[257.29841666666664,"median_c is shown on the screen, drawn."],[260.09641666666664,"centre_g is shown on the screen, written out."],[261.12941666666666,"centre_g is indicated — a transient flash."]]},{"start":262.30991666666665,"say":"The perpendicular bisectors meet at the circumcentre O, the centre of the circle through all three corners.","live":["plane","heading","triangle","vertex_a","vertex_b","vertex_c","mid_a","mid_b","mid_c","median_a","median_b","median_c","centre_g"],"does":[[262.30991666666665,"median_a is hidden from the screen."],[262.30991666666665,"median_b is hidden from the screen."],[262.30991666666665,"median_c is hidden from the screen."],[262.30991666666665,"mid_a is hidden from the screen."],[262.30991666666665,"mid_b is hidden from the screen."],[262.30991666666665,"mid_c is hidden from the screen."],[263.2854166666666,"bisector_a is shown on the screen, drawn."],[263.2854166666666,"bisector_b is shown on the screen, drawn."],[263.5354166666666,"bisector_c is shown on the screen, drawn."],[265.15441666666663,"centre_o is shown on the screen, written out."],[265.15441666666663,"centre_o is indicated — a transient flash."],[266.60541666666666,"circumcircle is shown on the screen, drawn."]]},{"start":269.34141666666665,"say":"The altitudes meet at the orthocentre H.","live":["plane","heading","triangle","vertex_a","vertex_b","vertex_c","centre_g","bisector_a","bisector_b","bisector_c","centre_o","circumcircle"],"does":[[269.34141666666665,"bisector_a is hidden from the screen."],[269.34141666666665,"bisector_b is hidden from the screen."],[269.34141666666665,"bisector_c is hidden from the screen."],[269.34141666666665,"circumcircle is hidden from the screen."],[269.8524166666666,"altitude_a is shown on the screen, drawn."],[269.8524166666666,"altitude_b is shown on the screen, drawn."],[270.1024166666666,"altitude_c is shown on the screen, drawn."],[271.5014166666666,"centre_h is shown on the screen, written out."],[271.5014166666666,"centre_h is indicated — a transient flash."]]},{"start":272.86741666666666,"say":"The construction lines can now leave.","live":["plane","heading","triangle","vertex_a","vertex_b","vertex_c","centre_g","centre_o","altitude_a","altitude_b","altitude_c","centre_h"],"does":[[272.86741666666666,"altitude_a is hidden from the screen."],[273.06741666666665,"altitude_b is hidden from the screen."],[273.26741666666663,"altitude_c is hidden from the screen."]]},{"start":275.9634166666666,"say":"Only three labelled points remain: the circumcentre O, the centroid G, and the orthocentre H.","live":["plane","heading","triangle","vertex_a","vertex_b","vertex_c","centre_g","centre_o","centre_h"],"does":[[279.22641666666664,"centre_o is indicated — a transient flash."],[280.4104166666666,"centre_g is indicated — a transient flash."],[282.0594166666666,"centre_h is indicated — a transient flash."]]},{"start":283.2799166666666,"say":"They lie on one straight line, exactly. This is the Euler line.","live":null,"does":[[283.2799166666666,"plane moves to a new place on the board."],[283.2799166666666,"euler_note is shown on the screen, written out."],[284.5514166666666,"euler_note (the \"collinear.\" part) is emphasized."],[284.9464166666666,"euler_line is shown on the screen, drawn."]]},{"start":288.9824166666666,"say":"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.","live":["euler_note","plane","heading","triangle","vertex_a","vertex_b","vertex_c","centre_g","centre_o","centre_h","euler_line"],"does":[[288.9824166666666,"euler_note (the \"collinear.\" part) is no longer emphasized."],[294.77641666666665,"triangle is redrawn as the numbers it depends on change."],[294.77641666666665,"vertex_c is redrawn as the numbers it depends on change."],[294.77641666666665,"centre_g is redrawn as the numbers it depends on change."],[294.77641666666665,"centre_o is redrawn as the numbers it depends on change."],[294.77641666666665,"centre_h is redrawn as the numbers it depends on change."],[294.77641666666665,"euler_line is redrawn as the numbers it depends on change."],[294.77641666666665,"centre_i is redrawn as the numbers it depends on change."],[294.77641666666665,"corner_x ticks to 5.3."],[294.77641666666665,"corner_y ticks to 6.5."],[295.67041666666665,"centre_o is indicated — a transient flash."],[296.00741666666664,"centre_g is indicated — a transient flash."],[296.48341666666664,"centre_h is indicated — a transient flash."]]},{"start":299.6079166666666,"say":"Move the corner back to the right, and the same line carries all three centres again.","live":null,"does":[[301.0884166666666,"triangle is redrawn as the numbers it depends on change."],[301.0884166666666,"vertex_c is redrawn as the numbers it depends on change."],[301.0884166666666,"centre_g is redrawn as the numbers it depends on change."],[301.0884166666666,"centre_o is redrawn as the numbers it depends on change."],[301.0884166666666,"centre_h is redrawn as the numbers it depends on change."],[301.0884166666666,"euler_line is redrawn as the numbers it depends on change."],[301.0884166666666,"centre_i is redrawn as the numbers it depends on change."],[301.0884166666666,"corner_x ticks to 6.6."],[301.0884166666666,"corner_y ticks to 6.0."],[302.5164166666666,"euler_line is indicated — a transient flash."]]},{"start":305.5369166666666,"say":"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.","live":null,"does":[[308.3294166666666,"ratio is shown on the screen, written out."],[308.3294166666666,"ratio (the \"O G\" part) is emphasized."],[308.3294166666666,"The segment (4.050000000000001, 3.6499999999999995) to (4.933333333333333, 3.4) in plane is lit up."],[309.15341666666666,"ratio (the \"1\" part) is emphasized."],[309.15341666666666,"ratio (the \"O G\" part) is no longer emphasized."],[310.29141666666663,"ratio (the \"1\" part) is no longer emphasized."],[310.29141666666663,"ratio (the \"G H\" part) is emphasized."],[310.29141666666663,"plane: retire a lit segment (unemphasize_line)."],[310.29141666666663,"The segment (4.933333333333333, 3.4) to (6.699999999999999, 2.900000000000001) in plane is lit up."],[310.9654166666666,"ratio (the \"2\" part) is emphasized."],[310.9654166666666,"ratio (the \"G H\" part) is no longer emphasized."],[314.80741666666665,"plane: retire a lit segment (unemphasize_line)."]]},{"start":315.4074166666666,"say":"The incentre I also moves with the triangle, but it does not generally lie on the Euler line.","live":["euler_note","ratio","plane","heading","triangle","vertex_a","vertex_b","vertex_c","centre_g","centre_o","centre_h","euler_line"],"does":[[315.4074166666666,"ratio (the \"2\" part) is no longer emphasized."],[316.45241666666664,"centre_i is shown on the screen, written out."],[316.45241666666664,"centre_i is indicated — a transient flash."],[318.82041666666663,"centre_i is indicated — a transient flash."]]},{"start":321.7429166666666,"say":"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.","live":["euler_note","ratio","plane","heading","triangle","vertex_a","vertex_b","vertex_c","centre_g","centre_o","centre_h","euler_line","centre_i"],"does":[[322.76441666666665,"centre_g is indicated — a transient flash."],[325.3424166666666,"centre_o is indicated — a transient flash."],[328.3954166666666,"centre_i is indicated — a transient flash."],[331.1934166666666,"centre_h is indicated — a transient flash."],[335.8834166666666,"euler_line is indicated — a transient flash."],[337.48345833333326,"euler_note is hidden from the screen — left the board."],[337.48345833333326,"heading is hidden from the screen — left the board."],[337.48345833333326,"plane is hidden from the screen — left the board."],[337.48345833333326,"triangle is hidden from the screen — plane left the board."],[337.48345833333326,"vertex_a is hidden from the screen — plane left the board."],[337.48345833333326,"vertex_b is hidden from the screen — plane left the board."],[337.48345833333326,"vertex_c is hidden from the screen — plane left the board."],[337.48345833333326,"centre_g is hidden from the screen — plane left the board."],[337.48345833333326,"centre_o is hidden from the screen — plane left the board."],[337.48345833333326,"centre_h is hidden from the screen — plane left the board."],[337.48345833333326,"euler_line is hidden from the screen — plane left the board."],[337.48345833333326,"centre_i is hidden from the screen — plane left the board."],[337.48345833333326,"ratio is hidden from the screen — left the board."]]}]}]},"durationSeconds":339,"chapters":[{"title":"Four Constructions","startSeconds":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. 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."},{"title":"When the Orthocentre Moves Outside","startSeconds":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? 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."},{"title":"The Euler Line","startSeconds":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. 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."}]}}