MathDB
0752

Source:

June 9, 2008
geometrycircumcircleincenteranalytic geometrygeometric transformationhomothetytrigonometry

Problem Statement

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