The Apollonius point of a reference triangle 
is the point
where the lines 
, 
, and 
concur. Here 
, 
, and 
are the points at which the enclosing
Apollonius circle 
is internally tangent
to the excircles 
, 
, and 
opposite 
, 
, and 
, respectively. The triangle 
is the Apollonius
triangle (Lozada 2025).
The Apollonius point has triangle center function
and is Kimberling center 
(Kimberling 1998, p. 102).
See also
Apollonius Circle,
Apollonius'
Problem,
Apollonius Triangle,
Excircles
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References
Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163-187, 1994.Kimberling,
C. "Triangle Centers and Central Triangles." Congr. Numer. 129,
1998.Kimberling, C. "Apollonius Point." https://faculty.evansville.edu/ck6/tcenters/recent/apollon.html.Kimberling,
C.; Iwata, S.; and Fukagawa, H. "Problem 1091 and Solution." Crux Math. 13,
128-129 and 217-218, 1987.Lozada, C. E. "Index of Triangles
Referenced in ETC." Updated Jan. 11, 2025. https://faculty.evansville.edu/ck6/encyclopedia/IndexOfTrianglesReferencedInETC.html.Referenced
on Wolfram|Alpha
Apollonius Point
Cite this as:
Weisstein, Eric W. "Apollonius Point."
From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ApolloniusPoint.html
Subject classifications
![alpha=sin^2Acos^2[1/2(B-C)]](/images/equations/ApolloniusPoint/NumberedEquation1.svg)
