The incentral circle is the circumcircle of the incentral triangle. It has radius R_I = sqrt(a b c f(a, b, c) f(b, c, a) f(c, a, b))/(8Δ(a + b)(a + c)(b + c)), where Δ is the area of the reference triangle and f(a, b, c) = a^3 - b a^2 + c a^2 - b^2 a - c^2 a - 3b c a + b^3 - c^3 - b c^2 + b^2 c. Its center function is a sixth-order polynomial that does not correspond to any Kimberling center. Its circle function is l = - (-a^3 + b^3 + c^3 + (-a + b + c)(a b + a c + b c))/(2(a + b)(a + c)(b + c)), corresponding to Kimberling center X_191.

anticomplementary triangle | central circle