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