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 $\omega$ 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 $\triangle PQR$ . Furthermore, QR is the polar of P, PQ is the polar of R, and PR is the polar of Q with respect to $\omega$ .^[1]^[4]^[5]

References

1 2 3 Chen, Evan (2016). Euclidean Geometry in Mathematical Olympiads. Mathematical Association of America. p. 179. ISBN 978-0883858394.
↑ Coxeter, H. S. M. (1987). Projective Geometry (2nd ed.). Springer-Verlag. ISBN 0-387-96532-7.
↑ Janković, Vladimir (2011). The IMO Compendium. Springer-Verlag. p. 15. ISBN 978-1-4419-9853-8.
↑ Bamberg, John (2023). Analytic Projective Geometry. Cambridge University Press. p. 208. ISBN 978-1-0092-6063-3.
↑ Heuristic ID Team (2021), HEURISTIC: For Mathematical Olympiad Approach 2nd Edition, p. 99. (in Indonesian)

External links

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[Chen-2016-1] 1 2 3 Chen, Evan (2016). Euclidean Geometry in Mathematical Olympiads. Mathematical Association of America. p. 179. ISBN 978-0883858394.

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

[3] Janković, Vladimir (2011). The IMO Compendium. Springer-Verlag. p. 15. ISBN 978-1-4419-9853-8.

[4] Bamberg, John (2023). Analytic Projective Geometry. Cambridge University Press. p. 208. ISBN 978-1-0092-6063-3.

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

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Brokard's theorem

Statement

See also

References

External links