Brocard points

In geometry, Brocard points are special points within a triangle. They are named after Henri Brocard (1845 – 1922), a French mathematician.

In a triangle ABC with sides a, b, and c, where the vertices are labeled A, B and C in counterclockwise order, there is exactly one point P such that the line segments AP, BP, and CP form the same angle, ω, with the respective sides c, a, and b, namely that

Point P is called the first Brocard point of the triangle ABC, and the angle ω is called the Brocard angle of the triangle. This angle has the property that

where ${\displaystyle \alpha ,\,\beta ,\,\gamma }$ are the vertex angles ${\displaystyle \angle CAB,\,\angle ABC,\,\angle BCA}$ respectively.

There is also a second Brocard point, Q, in triangle ABC such that line segments AQ, BQ, and CQ form equal angles with sides b, c, and a respectively. In other words, the equations ${\displaystyle \angle QCB=\angle QBA=\angle QAC}$ apply. Remarkably, this second Brocard point has the same Brocard angle as the first Brocard point. In other words, angle ${\displaystyle \angle PBC=\angle PCA=\angle PAB}$ is the same as ${\displaystyle \angle QCB=\angle QBA=\angle QAC.}$

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