MathDB
0723

Source:

April 28, 2008
geometryincentercircumcirclegeometric transformationhomothetytrigonometrysearch

Problem Statement

Let ABC ABC be a given triangle with the incenter I I, and denote by X X, Y Y, Z Z the intersections of the lines AI AI, BI BI, CI CI with the sides BC BC, CA CA, and AB AB, respectively. Consider Ka \mathcal{K}_{a} the circle tangent simultanously to the sidelines AB AB, AC AC, and internally to the circumcircle C(O) \mathcal{C}(O) of ABC ABC, and let A′ A^{\prime} be the tangency point of Ka \mathcal{K}_{a} with C \mathcal{C}. Similarly, define B′ B^{\prime}, and C′ C^{\prime}. Prove that the circumcircles of triangles AXA′ AXA^{\prime}, BYB′ BYB^{\prime}, and CZC′ CZC^{\prime} all pass through two distinct points.
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