Brocard circle

In

).

Equation

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

${\displaystyle b^{2}c^{2}x^{2}+a^{2}c^{2}y^{2}+a^{2}b^{2}z^{2}-a^{4}yz-b^{4}xz-c^{4}xy=0.}$

Related points

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]

Special cases

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

History

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]

References