Brocard point
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Brocard points are special points within a triangle. Henri Brocard was a French mathematician who lived from 1845 until 1922. He is best known for the following theorem.
In a triangle ABC with sides a, b, and c, 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. The following appies to this angle:
- cotω = cotα + cotβ + cotγ.
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 
apply. Remarkably, this second Brocard point has the same Brocard angle as the first Brocard point. In other words angle is the same as .
The two Brocard points are closely related to one another; In fact, the difference between the first and the second depends on the order in which the angles of triangle ABC are taken. So for example, the first Brocard point of triangle ABC is the same as the second Brocard point of triangle ACB.
The two Brocard points of a triangle ABC are isogonal conjugates of each other.
The most elegant construction of the Brocard points, in the following example the first Brocard point is presented, goes as follows:
Intersect the perpendicular bisector of side AB with the perpendicular to line BC through point B. Draw a circle with the intersection point as its center so that the circle goes through point B. This circle will also go through point A and intersect line BC at point B. We similarly construct a circle through points C and B, which touches line CA at point C, and a circle through points A and C, which touches line AB at point A. These three circles have a common point – the first Brocard point of triangle ABC!
The three circles just constructed are also designated as epicycles of triangle ABC. The second Brocard point is constructed in similar fashion.
