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tangent circles wanted, 2 right angles, 4 midpoints related

Source: 15th XMO (China) p1 https://artofproblemsolving.com/community/c3192772_geometry_regional_china

April 23, 2024
geometrytangent circles

Problem Statement

As shown in the figure, in the quadrilateral ABCDABCD, AB⊥BCAB\perp BC, AD⊥CDAD\perp CD, let EE be a point on line BDBD such that EC=CAEC = CA. The line perpendicular on lineAC AC passing through EE, intersects line ABAB at point FF, and line ADAD at point GG. Let XX and YY the midpoints of line segments AFAF and AGAG respectively. Let ZZ and WW be the midpoints of line segments BEBE and DEDE respectively. Prove that the circumscribed circle of △WBX\vartriangle WBX is tangent to the circumscribed circle of △ZDY\vartriangle ZDY. https://cdn.artofproblemsolving.com/attachments/0/3/1f6fca7509e6fd6cad662b42abd236fd4858ca.jpg
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