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# Equi-Brocard Center

There exists a triangulation point for which the triangles , , and have equal Brocard angles. This point is a triangle center known as the equi-Brocard center and is Kimberling center .

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

 (1) (2) (3)

and using the equation

 (4)

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

 (5)

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

Brocard Angle, First Brocard Point

## References

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." http://faculty.evansville.edu/ck6/encyclopedia/ETC.html#X368.

## Referenced on Wolfram|Alpha

Equi-Brocard Center

## Cite this as:

Weisstein, Eric W. "Equi-Brocard Center." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Equi-BrocardCenter.html