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| {{short description|Theorem about orthocenter and polars in circle geometry}} | {{short description|Theorem about orthocenter and polars in circle geometry}} | ||
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| '''Brokard's theorem''' is a theorem in ].<ref>{{cite book | '''Brokard's theorem''' is a theorem in ].<ref>{{cite book | ||
| | author = Coxeter, H. S. M. | | author = Coxeter, H. S. M. | ||
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| | edition = 2nd | | edition = 2nd | ||
| | year = 1987 | | year = 1987 | ||
| | publisher = Springer-Verlag | | publisher = ] | ||
| | isbn = 0-387-96532-7 | | isbn = 0-387-96532-7 | ||
| | pages = | | pages = | ||
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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>.<ref>Heuristic ID Team (2021), '''', p. 99. (in Indonesian)</ref></blockquote> | ||
| == See also == | == See also == | ||
| * ] | * ] | ||
| * ] | * ] | ||
| * ] | |||
| == References == | |||
| {{reflist}} | |||
| {{Reflist}} | |||
| == External link == | |||
| * | |||
| ] | ] | ||
| ] | ] | ||
Latest revision as of 20:37, 7 January 2025
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 polar 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 See also
References
- Coxeter, H. S. M. (1987). Projective Geometry (2nd ed.). Springer-Verlag. ISBN 0-387-96532-7.
- Heuristic ID Team (2021), HEURISTIC: For Mathematical Olympiad Approach 2nd Edition, p. 99. (in Indonesian)
External link
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