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Perpendicular bisector of secant tangent to circumcircle

Source: Iranian Geometry Olympiad 2016 Medium 5

May 26, 2017
geometryperpendicular bisectorcircumcircle

Problem Statement

Let the circles ω\omega and ω′\omega' intersect in points AA and BB. The tangent to circle ω\omega at AA intersects ω′\omega' at CC and the tangent to circle ω′\omega' at AA intersects ω\omega at DD. Suppose that the internal bisector of ∠CAD\angle CAD intersects ω\omega and ω′\omega' at EE and FF, respectively, and the external bisector of ∠CAD\angle CAD intersects ω\omega and ω′\omega' at XX and YY, respectively. Prove that the perpendicular bisector of XYXY is tangent to the circumcircle of triangle BEFBEF.
Proposed by Mahdi Etesami Fard
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