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Circles

Source: Bulgarian MO 2008, Day 1, Problem 1

May 17, 2008
geometrycircumcircletrigonometryangle bisectorgeometry proposed

Problem Statement

Let ABC ABC be an acute triangle and CL CL be the angle bisector of ∠ACB \angle ACB. The point P P lies on the segment CLCL such that \angle APB\equal{}\pi\minus{}\frac{_1}{^2}\angle ACB. Let k1 k_1 and k2 k_2 be the circumcircles of the triangles APC APC and BPC BPC. BP\cap k_1\equal{}Q, AP\cap k_2\equal{}R. The tangents to k1 k_1 at Q Q and k2 k_2 at B B intersect at S S and the tangents to k1 k_1 at A A and k2 k_2 at R R intersect at T T. Prove that AS\equal{}BT.
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