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McCay Circles Radical Circle

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McCayCirclesRadicalCircle

The radical circle of the McCay circles has center

alpha=-(5a^6-6b^2a^4-6c^2a^4-18b^2c^2a^2+2b^6+2c^6-3b^2c^4-3b^4c^2)/a,
(1)

which is not a Kimberling center, and radius

R_M=-((a^4-b^2a^2-c^2a^2+b^4+c^4-b^2c^2)sqrt(f(a,b,c)))/(9(a-b-c)(a+b-c)(a-b+c)(a+b+c)(a^2+b^2+c^2)),
(2)

where

f(a,b,c)=5a^6-3b^2a^4-3c^2a^4-3b^4a^2-3c^4a^2-51b^2c^2a^2+5b^6+5c^6-3b^2c^4-3b^4c^2.
(3)

Its circle function is

l=(g(a,b,c))/(9b(a-b-c)(a+b-c)c(a-b+c)(a+b+c)(a^2+b^2+c^2)),
(4)

where

g(a,b,c)=a^8-2b^2a^6-2c^2a^6+b^4a^4+c^4a^4-8b^2c^2a^4+3b^6a^2+3c^6a^2+16b^2c^4a^2+16b^4c^2a^2-b^8-c^8-5b^2c^6+10b^4c^4-5b^6c^2,
(5)

which is also does not correspond to any Kimberling center.

The McCay circles radical circle is real only in the case that the McCay circles do not intersect.

No Kimberling centers lie on it.

See also

McCay Circles, Radical Circle

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Cite this as:

Weisstein, Eric W. "McCay Circles Radical Circle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/McCayCirclesRadicalCircle.html

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