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2014 Geometry #7: 13-14-15 Circle

Source:

February 25, 2014
geometryanalytic geometrycircumcircleperpendicular bisector

Problem Statement

Triangle ABCABC has sides AB=14AB = 14, BC=13BC = 13, and CA=15CA = 15. It is inscribed in circle Γ\Gamma, which has center OO. Let MM be the midpoint of ABAB, let B′B' be the point on Γ\Gamma diametrically opposite BB, and let XX be the intersection of AOAO and MB′MB'. Find the length of AXAX.
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