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Circle Function


A central circle is a circle with trilinear equation

 (lalpha+mbeta+ngamma)(aalpha+bbeta+cgamma)+k(abetagamma+bgammaalpha+calphabeta)=0

such that l:m:n is a triangle center and k is a homogeneous function that is symmetric in the side lengths a, b, and c of the reference triangle.

In this work, the term "circle function" is used to refer to the function l corresponding to k=1. A triangle center X that is equal to l modulo factors symmetric in a, b, c and A, B, and C is then said to correspond to the circle function l. Circles with centers X corresponding to a given circle function l therefore have the same radical line with the circumcircle with l.

The circle function of a circle is related to the A-power (i.e., the circle power with respect to the A-vertex) by

 l=-(p_A)/(bc).

See also

Central Circle, Circle, Triangle Center

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Cite this as:

Weisstein, Eric W. "Circle Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CircleFunction.html

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