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== Statement == == Statement ==
<blockquote>'''Brokard's theorem'''. The points ''A'', ''B'', ''C'', and ''D'' lie in this order on a circle <math>\omega</math> 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 <math>\triangle PQR</math>. Furthermore, ''QR'' is the ] of ''P'', ''PQ'' is the polar of ''R'', and ''PR'' is the polar of ''Q'' with respect to <math>\omega</math>.<ref></ref>Heuristic ID Team (2021), ''HEURISTIC: For Mathematical Olympiad Approach 2nd Edition'', p. 99. (in Indonesian)</blockquote> <blockquote>'''Brokard's theorem'''. The points ''A'', ''B'', ''C'', and ''D'' lie in this order on a circle <math>\omega</math> 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 <math>\triangle PQR</math>. Furthermore, ''QR'' is the ] of ''P'', ''PQ'' is the polar of ''R'', and ''PR'' is the polar of ''Q'' with respect to <math>\omega</math>.<ref>Heuristic ID Team (2021), ''HEURISTIC: For Mathematical Olympiad Approach 2nd Edition'', p. 99. (in Indonesian)</ref></blockquote>


== See also == == See also ==

Revision as of 20:20, 7 January 2025

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.
  2. Heuristic ID Team (2021), HEURISTIC: For Mathematical Olympiad Approach 2nd Edition, p. 99. (in Indonesian)
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