Brocard point

There are two Brocard points within any triangle. The first Brocard point is the point $P$ in triangle $\triangle ABC$ labeled in counterclockwise order such that $\angle PAB=\angle PCA=\angle PBC$ , with the unique angle denoted as $\omega$ , the Brocard angle. The second Brocard point is a point $Q$ inside $\triangle ABC$ such that $\angle QAC=\angle QCA=\angle QBA$ . Moreover, these two triples of angles are equal. In general, we have $\angle PAB=\angle PCA=\angle PBC=\angle QAC=\angle QCA=\angle QBA=\omega$ The two Brocard points of a triangle only coincide when the triangle is equilateral. These points are named after French mathematician Henri Brocard.

The Brocard angle $\omega$ is given by the identities: $\begin{align*}\cot(\omega)&=\cot(A)+\cot(B)+\cot(C)\\&=\frac{a^2+b^2+c^2}{4\triangle}\end{align*}$ where $\triangle$ is the area of triangle $\triangle ABC$ with $\angle A, \angle B, \angle C$ opposite side $a, b, c$ , respectively.

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1999_AIME_Problems/Problem_14

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Brocard point

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