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Existence of point X and equal circles

Source: Iberoamerican Olympiad 2009, problem 3

September 23, 2009
geometryparallelogramsymmetrycircumcircle

Problem Statement

Let C1 C_1 and C2 C_2 be two congruent circles centered at O1 O_1 and O2 O_2, which intersect at A A and B B. Take a point P P on the arc AB AB of C2 C_2 which is contained in C1 C_1. AP AP meets C1 C_1 at C C, CB CB meets C2 C_2 at D D and the bisector of ∠CAD \angle CAD intersects C1 C_1 and C2 C_2 at E E and L L, respectively. Let F F be the symmetric point of D D with respect to the midpoint of PE PE. Prove that there exists a point X X satisfying \angle XFL \equal{} \angle XDC \equal{} 30^\circ and CX \equal{} O_1O_2.
Author: Arnoldo Aguilar (El Salvador)
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