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 \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.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

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