Brokard's theorem
{{short description|Theorem about orthocenter and polars in circle geometry}}
Brokard's theorem (also known as Brocard's theorem{{cite book
| last = Chen
| first = Evan
| title = Euclidean Geometry in Mathematical Olympiads
| publisher = Mathematical Association of America
| year = 2016
| isbn = 978-0883858394
| page = 179
}}) is a theorem in projective geometry.{{cite book
| author = Coxeter, H. S. M.
| author-link = H. S. M. Coxeter
| title = Projective Geometry
| edition = 2nd
| year = 1987
| publisher = Springer-Verlag
| isbn = 0-387-96532-7
| pages =
| no-pp = true}} It is commonly used in Olympiad mathematics.{{cite book |last=Janković |first=Vladimir |title=The IMO Compendium |publisher=Springer-Verlag |year= 2011 |isbn=978-1-4419-9853-8 |page=15}} 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 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 .Heuristic ID Team (2021), [https://books.google.com.br/books/about/HEURISTIC_For_Mathematical_Olympiad_Appr.html?id=UVpXEAAAQBAJ&redir_esc=y HEURISTIC: For Mathematical Olympiad Approach 2nd Edition], p. 99. (in Indonesian)
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.{{cite book |last=Bamberg |first=John |title=Analytic Projective Geometry |publisher=Cambridge University Press |year=2023 |isbn=978-1-0092-6063-3 |page=208}}
See also
References
{{Reflist}}
External link
- [https://artofproblemsolving.com/community/c90445h1192060_brokards_theorem__proof_without_words?srsltid=AfmBOopPmJTgYjsi-wfCfmrx63mj0QDsbZ4EYVSN8xyNeP8cl5uYWvJA A proof without words of Brokard's theorem]
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