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Circle constructed from a triangle From Wikipedia, the free encyclopedia
In geometry, the Brocard circle (or seven-point circle) is a circle derived from a given triangle. It passes through the circumcenter and symmedian point of the triangle, and is centered at the midpoint of the line segment joining them (so that this segment is a diameter).
In terms of the side lengths , , and of the given triangle, and the areal coordinates for points inside the triangle (where the -coordinate of a point is the area of the triangle made by that point with the side of length , etc), the Brocard circle consists of the points satisfying the equation[1]
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]
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