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circle (X,XA) tangent to FB, sqaure and equilateral related

Source: (2021-) 2022 XV 15th Dürer Math Competition Finals Day 1 E1

November 29, 2022
geometrytangentEquilateralSqaure

Problem Statement

To the exterior of side ABAB of square ABCDABCD, we have drawn the regular triangle ABEABE. Point AA reflected on line BEBE is FF, and point EE reflected on line BFBF is GG. Let the perpendicular bisector of segment FGFG meet segment ADAD at XX. Show that the circle centered at XX with radius XAXA touches lineFB FB.
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