# Coxeter- Theorem 9.22 - Figure 9.2B

- Author:
- Pinyi Yao

- Topic:
- Geometry

The dual of 9.13 was discovered independently by William Braikenridge and Colin Maclaurin, about 1733:
Theorem 9.22: If the sides of a variable triangle pass through three fixed non-collinear points P, Q, R, while two vertices lie on fixed lines a and b,, not concurrent with PQQ, then the third vertex describes a conic.
Construction of figure 9.2B: Starting with three fixed points P, Q, R (in color blue), and two fixed lines a, b (in color blue). Then draw a random line through P (line x', keep the other point used to draw this line in order to move this line around later), we get F=x'·b. Connect FR, we get G=y'·a. Connect QG, we get H=x'·y'. HFG is the variable triangle, and H is the third vertex. Drag point E around, you can see the locus of point H is a conic.
This enables us to locate any number of points on the conic through five given points.