Brokard's theorem (projective geometry)

Brokard's theorem is a theorem in projective geometry. 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 $\omega$ 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 $\triangle PQR$. Furthermore, QR is the polar of P, PQ is the polar of R, and PR is the polar of Q with respect to $\omega$.

See also

• Orthocenter
• Power of a point
• Projective geometry