Brokard's theorem
Brokard's theorem (also known as Brocard's theorem[1]) is a theorem in projective geometry.[2] It is commonly used in Olympiad mathematics.[1][3] It is named after French mathematician Henri Brocard.
Statement
[edit]Brokard's theorem. The points A, B, C, and D lie in this order on a circle 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 . Furthermore, QR is the polar of P, PQ is the polar of R, and PR is the polar of Q with respect to .[4][1]
An equivalent formulation of Brokard's theorem states that the orthocenter of the diagonal triangle of a cyclic quadrilateral is the circumcenter of the cyclic quadrilateral.[5]
See also
[edit]References
[edit]- ^ a b c 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.
- ^ Heuristic ID Team (2021), HEURISTIC: For Mathematical Olympiad Approach 2nd Edition, p. 99. (in Indonesian)
- ^ Bamberg, John (2023). Analytic Projective Geometry. Cambridge University Press. p. 208. ISBN 978-1-0092-6063-3.
External link
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