Brocard circle explained

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).

Equation

In terms of the side lengths

a , b , and c of the given triangle, and the areal coordinates (x,y,z) for points inside the triangle (where the x -coordinate of a point is the area of the triangle made by that point with the side of length a , etc), the Brocard circle consists of the points satisfying the equationb ^{2} c ^{2} x ^{2} + a ^{2} c ^{2} y ^{2} + a ^{2} b ^{2} z ^{2} - a ^{4} y z - b ^{4} x z - c ^{4} x y =0 .

Related points

The two Brocard points lie on this circle, as do the vertices of the Brocard triangle.^[1] 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.^[2]

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,^[3] who presented a paper on it to the French Association for the Advancement of Science in Algiers in 1881.

See also

• Nine-point circle

Notes and References

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