Brokard's theorem (also known as Brocard's theorem) is a theorem in

projective geometry In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting (''p ...

. It is commonly used in Olympiad mathematics. It is named after French mathematician

Henri Brocard Pierre René Jean Baptiste Henri Brocard (; 12 May 1845 – 16 January 1922) was a French meteorologist and mathematician, in particular a geometer. His best-known achievement is the invention and discovery of the properties of the Brocard p ...

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 Polar may refer to: Geography * Geographical pole, either of the two points on Earth where its axis of rotation intersects its surface ** Polar climate, the climate common in polar regions ** Polar regions of Earth, locations within the polar circ ...
of ''P'', ''PQ'' is the polar of ''R'', and ''PR'' is the polar of ''Q'' with respect to $\omega$ .

An equivalent formulation of Brokard's theorem states that the

orthocenter The orthocenter of a triangle, usually denoted by , is the point (geometry), point where the three (possibly extended) altitude (triangle), altitudes intersect. The orthocenter lies inside the triangle if and only if the triangle is acute trian ...

of the diagonal triangle of a

cyclic quadrilateral In geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral (four-sided polygon) whose vertex (geometry), vertices all lie on a single circle, making the sides Chord (geometry), chords of the circle. This circle is called ...

is the

circumcenter In geometry, the circumscribed circle or circumcircle of a triangle is a circle that passes through all three vertices. The center of this circle is called the circumcenter of the triangle, and its radius is called the circumradius. The circumcen ...

of the cyclic quadrilateral.

References

External link

A proof without words of Brokard's theorem
Projective geometry Theorems in geometry {{geometry-stub

Statement

See also

References

External link