Brokard's theorem

Brokard's theorem (also known as Brocard's theorem ^[1] ) is a theorem in projective geometry. ^[2] It is commonly used in Olympiad mathematics. ^{[1] [3]} It is named after French mathematician Henri Brocard.

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 ${\displaystyle \omega }$. ^{[4] [1]}

An equivalent formulation of Brokard's theorem states that the orthocenter of the diagonal triangle of a cyclic quadrilateral is the circumcenter of the cyclic quadrilateral. ^[5]

See also

• Orthocenter
• Power of a point
• Pole and polar

References

1. Chen, Evan (2016). Euclidean Geometry in Mathematical Olympiads. Mathematical Association of America. p. 179. ISBN   978-0883858394.
2. Coxeter, H. S. M. (1987). Projective Geometry (2nd ed.). Springer-Verlag. ISBN   0-387-96532-7.
3. Janković, Vladimir (2011). The IMO Compendium. Springer-Verlag. p. 15. ISBN   978-1-4419-9853-8.
4. Heuristic ID Team (2021), HEURISTIC: For Mathematical Olympiad Approach 2nd Edition , p. 99. (in Indonesian)
5. Bamberg, John (2023). Analytic Projective Geometry. Cambridge University Press. p. 208. ISBN   978-1-0092-6063-3.

External links

• A proof without words of Brokard's theorem