Brocard circle

In terms of the side lengths ${\displaystyle a}$, ${\displaystyle b}$, and ${\displaystyle c}$ of the given triangle, and the areal coordinates ${\displaystyle (x,y,z)}$ for points inside the triangle (where the ${\displaystyle x}$-coordinate of a point is the area of the triangle made by that point with the side of length ${\displaystyle a}$, etc), the Brocard circle consists of the points satisfying the equation[1]

${\displaystyle a^{2}yz+b^{2}zx+c^{2}xy={\frac {a^{2}b^{2}c^{2}(x+y+z)}{a^{2}+b^{2}+c^{2}}}\left({\frac {x}{a^{2}}}+{\frac {y}{b^{2}}}+{\frac {z}{c^{2}}}\right).}$

The two Brocard points lie on this circle, as do the vertices of the Brocard triangle.[2] These five points, together with the other two points on the circle (the circumcenter and symmedian), justify the name "seven-point circle".

The Brocard circle is concentric with the first Lemoine circle.[3]

If the triangle is equilateral, the circumcenter and symmedian coincide and therefore the Brocard circle reduces to a single point.[4]

The Brocard circle is named for Henri Brocard,[5] who presented a paper on it to the French Association for the Advancement of Science in Algiers in 1881.[6]

• Weisstein, Eric W. "Brocard Circle". MathWorld.

• Nine-point circle