| Revision as of 18:36, 26 December 2024 editGregariousMadness (talk | contribs)Extended confirmed users1,324 edits ←Created page with '{{short description|Theorem about orthocenter and polars in circle geometry}} '''Brokard's theorem''' is a theorem in projective geometry.<ref>{{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 = Theorem 9.15, p. 83 | no-pp = true}}</ref> It is commonly used in Olympiad mathematics. == Statement ==...' | Revision as of 18:37, 26 December 2024 edit undoGregariousMadness (talk | contribs)Extended confirmed users1,324 edits →StatementNext edit → | ||
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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>.</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>.</blockquote> | ||
| == See also == | == See also == | ||
Revision as of 18:37, 26 December 2024
Theorem about orthocenter and polars in circle geometryBrokard'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 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 of P, PQ is the polar of R, and PR is the polar of Q with respect to .
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 of P, PQ is the polar of R, and PR is the polar of Q with respect to See also
- Coxeter, H. S. M. (1987). Projective Geometry (2nd ed.). Springer-Verlag. Theorem 9.15, p. 83. ISBN 0-387-96532-7.
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