Let A′ be an arbitrary point on the side BC of a triangle ABC. Denote by TAb, TAc the circles simultanously tangent to AA′, A′B, Γ and AA′, A′C, Γ, respectively, where Γ is the circumcircle of ABC. Prove that TAb, TAc are congruent if and only if AA′ passes through the Nagel point of triangle ABC.
(If M,N,P are the points of tangency of the excircles of the triangle ABC with the sides of the triangle BC, CA and AB respectively, then the Nagel point of the triangle is the intersection point of the lines AM, BN and CP.) geometrycircumcircleincenteranalytic geometrygeometric transformationhomothetytrigonometry