Brokard's theorem (also known as Brocard's theorem) is a theorem on poles and polars in projective geometry commonly used in Olympiad mathematics. 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 ω {\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 △ P Q R {\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 ω {\displaystyle \omega } .
See also Orthocenter Power of a point Pole and polar
References
External links Proof of Brokard's theorem A proof without words of Brokard's theorem