WEBVTT 00:00:00.035 --> 00:00:02.799 A triangle has more than corners and sides. 00:00:02.995 --> 00:00:06.302 Four different geometric rules locate four special points, 00:00:06.385 --> 00:00:10.694 and the surprise is that each rule makes three independently defined lines meet. 00:00:12.206 --> 00:00:17.464 The four destinations are the centroid G, the circumcentre O, the incentre I, 00:00:17.547 --> 00:00:19.426 and the orthocentre H. 00:00:19.509 --> 00:00:20.657 Their rules are different, 00:00:20.740 --> 00:00:24.305 so their dots occupy different places in the same triangle. 00:00:25.101 --> 00:00:28.002 The first rule begins at the midpoint of every side. 00:00:28.085 --> 00:00:32.324 Those three landing places are A prime, B prime, and C prime. 00:00:33.236 --> 00:00:36.825 A segment from a corner to the opposite midpoint is a median. 00:00:37.021 --> 00:00:41.886 The median from A lands at A prime, and the median from B lands at B prime. 00:00:42.683 --> 00:00:43.970 Those two meet. 00:00:44.053 --> 00:00:48.686 The third median, from C to C prime, passes through the very same point. 00:00:49.138 --> 00:00:51.705 That common point is the centroid G. 00:00:52.130 --> 00:00:54.775 The centroid is the triangle's balance point. 00:00:54.858 --> 00:00:59.257 Along the median from A, the length A G is two thirds of the whole median, 00:00:59.340 --> 00:01:01.895 while G A prime is one third. 00:01:02.091 --> 00:01:05.110 The two lengths therefore have ratio two to one. 00:01:06.023 --> 00:01:08.366 A second rule starts from the same midpoints, 00:01:08.449 --> 00:01:10.424 but turns through a right angle. 00:01:10.621 --> 00:01:14.324 A perpendicular bisector crosses a side square on and divides it 00:01:14.324 --> 00:01:15.985 into equal halves. 00:01:16.898 --> 00:01:19.070 Choose any point on that line. 00:01:19.266 --> 00:01:24.666 Its distance to B equals its distance to C, the two ends of the side it bisects. 00:01:25.462 --> 00:01:29.840 The other two perpendicular bisectors carry the same promise for their sides. 00:01:30.037 --> 00:01:33.729 All three pass through one point, the circumcentre O. 00:01:34.340 --> 00:01:40.378 Because O lies on all three bisectors, O A, O B, and O C are equal. 00:01:40.690 --> 00:01:42.909 Call their common length R. 00:01:43.392 --> 00:01:48.603 A circle with centre O and radius R therefore passes through A, B, and C. 00:01:48.686 --> 00:01:53.841 This is the circumcircle, and the gap from O back to G confirms that the circumcentre 00:01:53.841 --> 00:01:55.955 and centroid are different points. 00:01:56.659 --> 00:01:59.106 The third rule belongs to the corners. 00:01:59.189 --> 00:02:03.312 An angle bisector splits one corner into two equal angles. 00:02:04.108 --> 00:02:08.091 The angle bisectors from A and C meet the first at one point. 00:02:08.195 --> 00:02:10.134 This is the incentre I. 00:02:10.513 --> 00:02:14.656 The shortest distance from I to side A B is a perpendicular drop. 00:02:14.739 --> 00:02:19.663 The corresponding drops to B C and C A have exactly the same length. 00:02:19.975 --> 00:02:24.399 Their matching marks record the equality, and that common length is r. 00:02:24.824 --> 00:02:29.571 The circle centred at I with radius r reaches every side and crosses none. 00:02:29.654 --> 00:02:34.635 It touches A B, B C, and C A at the three perpendicular feet. 00:02:34.948 --> 00:02:36.888 This is the incircle. 00:02:38.102 --> 00:02:41.389 The fourth rule returns to perpendicular lines. 00:02:41.585 --> 00:02:46.392 An altitude passes through a corner and meets the opposite side at a right angle. 00:02:47.305 --> 00:02:51.625 The altitudes from B and C pass through the first altitude at one point. 00:02:51.728 --> 00:02:55.050 This fourth meeting point is the orthocentre H. 00:02:55.846 --> 00:03:00.967 For this acute triangle, every altitude foot and the orthocentre lie inside. 00:03:01.163 --> 00:03:04.113 An obtuse angle changes that geometry. 00:03:05.159 --> 00:03:10.547 At R, the angle is greater than a right angle, so this triangle is obtuse. 00:03:10.743 --> 00:03:14.819 The question is still the same: where do its three altitudes meet? 00:03:15.616 --> 00:03:18.248 Continue side Q R beyond R. 00:03:18.332 --> 00:03:22.814 The perpendicular from P meets that extended line outside the triangle. 00:03:23.518 --> 00:03:26.886 Continue side R P beyond R as well. 00:03:27.082 --> 00:03:31.936 The perpendicular from Q has its own right-angle foot beyond the triangle. 00:03:32.733 --> 00:03:36.178 The altitude from R meets P Q on the side itself. 00:03:36.262 --> 00:03:38.535 Its right-angle mark is inside, 00:03:38.619 --> 00:03:42.346 but the line continues upward to meet the other two altitudes. 00:03:43.143 --> 00:03:45.884 All three altitudes still meet at one point. 00:03:46.080 --> 00:03:49.912 The orthocentre H lies outside an obtuse triangle. 00:03:50.616 --> 00:03:52.904 The defining rule did not fail. 00:03:53.101 --> 00:03:57.896 Extending the two short sides exposed the same concurrency beyond the triangle. 00:03:59.028 --> 00:04:03.359 The four centres return on a triangle whose top corner can move. 00:04:03.556 --> 00:04:07.295 First, the three side midpoints mark where the medians land. 00:04:08.347 --> 00:04:10.632 The median from A reaches A prime, 00:04:10.716 --> 00:04:15.964 the median from B reaches B prime, and the median from C reaches C prime. 00:04:16.579 --> 00:04:19.308 Their meeting point is the centroid G. 00:04:20.360 --> 00:04:23.631 The perpendicular bisectors meet at the circumcentre O, 00:04:23.715 --> 00:04:26.572 the centre of the circle through all three corners. 00:04:27.369 --> 00:04:30.260 The altitudes meet at the orthocentre H. 00:04:30.685 --> 00:04:33.113 The construction lines can now leave. 00:04:33.910 --> 00:04:39.062 Only three labelled points remain: the circumcentre O, the centroid G, 00:04:39.146 --> 00:04:41.062 and the orthocentre H. 00:04:41.963 --> 00:04:45.052 They lie on one straight line, exactly. 00:04:45.353 --> 00:04:47.270 This is the Euler line. 00:04:48.298 --> 00:04:52.340 That alignment belongs to the triangle, not to this one shape. 00:04:52.629 --> 00:04:58.598 As the top corner moves left, O, G, and H move with it and remain collinear. 00:04:59.360 --> 00:05:04.190 Move the corner back to the right, and the same line carries all three centres again. 00:05:04.952 --> 00:05:07.286 Their spacing is fixed as well. 00:05:07.448 --> 00:05:14.055 O G is one part, while G H is two parts, so O G to G H is one to two. 00:05:14.852 --> 00:05:18.811 The incentre I also moves with the triangle, but it does not generally lie 00:05:18.811 --> 00:05:20.309 on the Euler line. 00:05:21.106 --> 00:05:23.402 The centroid G comes from medians, 00:05:23.486 --> 00:05:26.629 the circumcentre O from perpendicular bisectors, 00:05:26.713 --> 00:05:31.950 the incentre I from angle bisectors, and the orthocentre H from altitudes. 00:05:32.147 --> 00:05:36.153 Four independent rules reveal a remarkably organised triangle.