Coxeter- Theorem 9.15 - Figure 9.1E (left)
- Author:
- Pinyi Yao
- Topic:
- Geometry
Theorem 9.15: Brianchon's Theorem: If a hexagon is circumscribed about a conic the three diagonals are concurrent.
Proof: In Figure 9.1D, we see a hexagon ABCYXE whose six sides all touch a conic. The three lines AY, BX, CE, which join pairs of opposite vertices, are naturally called diagonals of the hexagon. Theorem 9.14 tells us that, if the diagonals of a hexagon are concurrent, the six sides all touch a conic. Conversely, if all the sides of a hexagon touch a conic, five of them can be identified with the lines DE,EA,AB,BC,CD. Since the given conic is the only one that touches these fixed lines, the sixth side must coincide with one of the lines XY for which BX·AY lies on CE.
Figure 9.1E illustrates this in a more natural notation: the Brianchon hexagon is ABCDEF and its diagonals are AD, BE, CF.
Construction of 9.1E: Construct a conic with any five points A1,B1,C1,D1,E1. Choose a random point F1 on the conic. Then draw all the tangent lines of the six points, we get six intersection points A,B,C,D,E,F, and it is our hexagon ABCDEF. Connect AD, BE, CF, they are concurrent at a point G. Move point F1 around, you can see the three diagonals are still concurrent.