Brocard circle


In geometry, the Brocard circle for a triangle is a circle defined from a given triangle. It passes through the circumcenter and symmedian of the triangle, and is centered at the midpoint of the line segment joining them.

Equation

In terms of the side lengths,, and of the given triangle, and the areal coordinates for points inside the triangle, the Brocard circle consists of the points satisfying the equation

Related points

The two Brocard points lie on this circle, as do the vertices of the Brocard triangle.
These five points, together with the other two points on the circle, justify the name "seven-point circle".
The Brocard circle is concentric with the first Lemoine circle.

Special cases

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

History

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