Brokard's theorem

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 $\omega$ 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 $\mathrm{△}PQR$. 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
• Pole and polar

External links

• Proof of Brokard's theorem
• A proof without words of Brokard's theorem

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