F3 · Chapter 6
Special Lines and Centres in a Triangle
A triangle has four special centres. Each one is where a family of special lines meet. The Cantonese exam chant 四心經 helps you pair each centre with its lines — and with the circle, when there is one.
《四心經》
岳垂心orthocentre
父
教
我
攞
高高線altitude
分
身形心centroid
在
試
場
滅中線median
敵
人
耐內心incentre
心
分角平分線angle bisector
割
細
圓內切圓inscribed circle
圈
開外心circumcentre
心
中垂直平分線perp. bisector
分
大外接圓circumcircle
考
完
耐 is read as a hook for 內; 分割 for the idea of bisecting an angle; 中分 for “bisect in the middle” (a perpendicular bisector). The terms inscribed circle and circumcircle appear in the chant, but they are not required in the F3 curriculum — the textbook treats them as optional names for the circles determined by the incentre and circumcentre.
| Chant cue | Centre | Special line(s) | Related circle |
|---|---|---|---|
| 岳父教我攞高分 | 垂心 H · orthocentre | 高線 · altitude | — |
| 身在試場滅敵人 | 形心 G · centroid | 中線 · median | — |
| 耐心分割細圓圈 | 內心 I · incentre | 角平分線 · angle bisector | 內切圓 · inscribed circle (term not required) |
| 開心中分大考完 | 外心 O · circumcentre | 垂直平分線 · perpendicular bisector | 外接圓 · circumcircle (term not required) |
Topic pages
Follow the textbook order. Green-card properties and their proofs sit inside each topic. Pages marked Non-foundation follow §6.2–6.3 of the book.
0Quick Review — isosceles, congruence and similarity
6.1AAngle bisectors — property and converse
6.1BPerpendicular bisectors — property and converse
6.2Incentre I — three angle bisectors
6.2CCircumcentre O — three perpendicular bisectors
6.3AMedians and altitudes
6.3BCentroid G and the 2 : 1 property
6.3BOrthocentre H
MoreSpecial triangles — isosceles and equilateral