Consider the excircles J_A, J_B, and J_C of a triangle, and the external Apollonius circle Γ tangent externally to all three. Denote the contact point of Γ and J_A by A', etc. Then the lines A A', B B', and C C' concur in a point known as the Apollonius point, which has triangle center function α = sin^2 A cos^2[1/2(B - C)] and is Kimberling center X_181.

Apollonius circle | Apollonius' problem | excircles

Apollonius of Perga