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Mixtilinear Incircles Radical Circle

MixtilinearIncirclesRadicalCircle

The radical circle of the mixtilinear incircles has a center with trilinear center function

alpha_(999)=cosA-2,
(1)

which is Kimberling center X_(999). It has radius

r=1/(f(a,b,c))sqrt(-(3(-a+b+c)(a^4b^2c^2-a^2b^4c^2+2a^2b^3c^3-a^2b^2c^4))/((a+b+c))),
(2)

where

f(a,b,c)=a^3-a^2b-ab^2+b^3-a^2c+6abc-b^2c-ac^2-bc^2+c^3.
(3)

Since the argument of the square root is always negative, the radical circle is therefore always imaginary. As a result, no Kimberling centers lie on it.

It has circle function

l=(4abc(a-b-c))/((a+b+c)(a^3-a^2b-ab^2+b^3-a^2c+6abc-b^2c-ac^2-bc^2+c^3)),
(4)

corresponding to Kimberling center X_9.

See also

Mixtilinear Circle, Mixtilinear Incircles, Radical Circle

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Cite this as:

Weisstein, Eric W. "Mixtilinear Incircles Radical Circle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MixtilinearIncirclesRadicalCircle.html

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