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Equi-Brocard Center


EquiBrocardCenter

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

a^'=(bcsqrt(beta^2+gamma^2+2betagammacosA))/((aalpha+bbeta+cgamma))
(1)
b^'=(acsqrt(alpha^2+beta^2+2alphabetacosB))/((aalpha+bbeta+cgamma))
(2)
c^'=(absqrt(alpha^2+beta^2+2alphabetacosC))/((aalpha+bbeta+cgamma))
(3)

and using the equation

 cotomega=(a^2+b^2+c^2)/(4Delta),
(4)

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

 (a^2+b^('2)+c^('2))/(Delta_(ab^'c^'))=(a^('2)+b^2+c^('2))/(Delta_(a^'bc^'))=(a^('2)+b^('2)+c^2)/(Delta_(a^'b^'c)),
(5)

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

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

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