Brokard's theorem

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

$\mathrm{△}PQR$

. 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 link

• A proof without words of Brokard's theorem