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Apollonius Point


ApolloniusPoint

Consider the excircles J_A, J_B, and J_C of a triangle, and the external Apollonius circle Gamma tangent externally to all three. Denote the contact point of Gamma and J_A by A^', etc. Then the lines AA^', BB^', and CC^' concur in a point known as the Apollonius point, which has triangle center function

 alpha=sin^2Acos^2[1/2(B-C)]

and is Kimberling center X_(181) (Kimberling 1998, p. 102).


See also

Apollonius Circle, Apollonius' Problem, Excircles

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References

Kimberling, C. "Apollonius Point." http://faculty.evansville.edu/ck6/tcenters/recent/apollon.html.Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163-187, 1994.Kimberling, C.; Iwata, S.; and Hidetosi, F. "Problem 1091 and Solution." Crux Math. 13, 128-129 and 217-218, 1987.

Referenced on Wolfram|Alpha

Apollonius Point

Cite this as:

Weisstein, Eric W. "Apollonius Point." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ApolloniusPoint.html

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