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Reflection Circle


ReflectedCircle

The reflection circle, a term coined here for the first time, is the circumcircle of the reflection triangle. It has center at Kimberling center X_(195), which is the X_5-Ceva conjugate of X_3.

The radius is given by

R_R=1/(S^3(OH^2-4R^2))product_(cyclic)sqrt(S^2(OH^2+2S_A)+2S_AS_BS_C)
(1)
=(sqrt(f(a,b,c)f(b,c,a)f(c,a,b)))/(4Delta|a^6-b^2a^4-c^2a^4-b^4a^2-c^4a^2-b^2c^2a^2+b^6+c^6-b^2c^4-b^4c^2|),
(2)

where O is the circumcenter, H is the orthocenter, S, S_A, S_B, and S_C is Conway triangle notation, Delta is the area of the reference triangle, and

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

Its circle function interestingly corresponds to the same triangle center as its center: X_(195).

Its A-power is given by

 p_A=(8S^4-b^2c^2(3S^2+S_BS_C))/(2(OH^2-4R^2)S^2)
(4)

(P. Moses, pers. comm., Feb. 3, 2005).

No Kimberling centers lie on it. However, the anticomplements of X_(110) and X_(930) lie on it, as do the reflection of X_(146) in X_4 and X_3 in X_(1263) (P. Moses, pers. comm., Jan. 31, 2005).


See also

Central Circle, Reflection Triangle

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Cite this as:

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

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