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Theorem about orthocenter and polars in circle geometry
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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 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

References

  1. Coxeter, H. S. M. (1987). Projective Geometry (2nd ed.). Springer-Verlag. ISBN 0-387-96532-7.
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