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Symmedial Circle

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SymmedialCircle

The symmedial circle is the circumcircle of the symmedial triangle. It has circle function

l=(bc(a^4-a^2b^2-b^4-a^2c^2-b^2c^2-c^4))/(2(a^2+b^2)(a^2+c^2)(b^2+c^2)),
(1)

which does not correspond to any Kimberling center, and radius

R_S=(sqrt(f(a,b,c)f(b,c,a)f(c,a,b)))/(2(a^2+b^2)(b^2+c^2)(c^2+a^2))R,
(2)

where R is the circumradius of the reference triangle and

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

Its center has trilinear center function

alpha=a(a^6b^2-2a^2b^6+b^8+a^6c^2-6a^4b^2c^2-3a^2b^4c^2+b^6c^2 -3a^2b^2c^4-6b^4c^4-2a^2c^6+b^2c^6+c^8),
(4)

which also does not correspond to any Kimberling center, but lies on the line L(X_(185),X_(585)) (P. Moses, pers. comm., Jan. 7, 2005).

The symmedial circle passes through X_(125), the center of the Jerabek hyperbola.

See also

Symmedial Triangle

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Cite this as:

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

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