Equi-Brocard Center


There exists a triangulation point Y for which the triangles BYC, CYA, and AYB have equal Brocard angles. This point is a triangle center known as the equi-Brocard center and is Kimberling center X_(368).

It has a complicated triangle center function given by the unique positive real root of a tenth-order polynomial f(a,b,c) in alpha, which is actually fifth-order in alpha^2. The polynomial can be found by computing the distances from each of the vertices to the triangulation point


and using the equation


where omega is the Brocard angle and Delta is the triangle area to obtain the three equations


where Delta_(xyz) is the area of the triangle with side lengths x, y, and z (which can be computed using Heron's formula).

See also

Brocard Angle, First Brocard Point

Explore with Wolfram|Alpha


Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163-187, 1994.Kimberling, C. "Clark Kimberling's Encyclopedia of Triangle Centers--ETC."

Referenced on Wolfram|Alpha

Equi-Brocard Center

Cite this as:

Weisstein, Eric W. "Equi-Brocard Center." From MathWorld--A Wolfram Web Resource.

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