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Tricky geo problem

Source: 2008 Korea National

June 9, 2015
geometry

Problem Statement

Points A,B,C,D,EA,B,C,D,E lie in a counterclockwise order on a circle OO, and AC=CEAC = CE P=BD∩ACP=BD \cap AC, Q=BD∩CEQ=BD \cap CE Let O1O_1 be the circle which is tangent to AP‾,BP‾\overline {AP}, \overline {BP} and arc ABAB (which doesn't contain CC) Let O2O_2 be the circle which is tangent DQ‾,EQ‾\overline {DQ}, \overline {EQ} and arc DEDE (which doesn't contain CC) Let O1∩O=R,O2∩O=S,RP∩QS=XO_1 \cap O = R, O_2 \cap O = S, RP \cap QS = X Prove that XCXC bisects ∠ACE\angle ACE
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