Brokard's theorem

Brokard's theorem (also known as Brocard's theorem^[1]) is a theorem on poles and polars in projective geometry commonly used in Olympiad mathematics.^[1][2][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