Brokard's theorem

Brokard's theorem is a theorem in projective geometry. ^[1] It is commonly used in olympiad mathematics.

Statement

Brokard's theorem. The points A, B, C, and D lie in this order on a circle ${\displaystyle \omega }$ with center O. Lines AC and BD intersect at P, AB and DC intersect at Q, and AD and BC intersect at R. Then O is the orthocenter of ${\displaystyle \triangle PQR}$. Furthermore, QR is the polar of P, PQ is the polar of R, and PR is the polar of Q with respect to ${\displaystyle \omega }$. ^[2]

See also

• Orthocenter
• Power of a point
• Pole and polar

References

1. Coxeter, H. S. M. (1987). Projective Geometry (2nd ed.). Springer-Verlag. ISBN   0-387-96532-7.
2. Heuristic ID Team (2021), HEURISTIC: For Mathematical Olympiad Approach 2nd Edition , p. 99. (in Indonesian)

External link

• A proof without words of Brokard's theorem