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Perpendiculars to AD and tangent circles

Source: Moldova TST 2024 P2

June 9, 2024
geometry

Problem Statement

In the acute-angled triangle ABCABC, let ADAD, D∈BCD \in BC be the AA-angle bisector. The perpenducular to BCBC through DD and the perpendicular to ADAD through AA meet at II. The circle with center II and radius IDID, intersects ABAB and ACAC at FF and EE respectively. On the arc FEFE, which does not contain AA, of the circumcircle of AFEAFE, consider a point XX, such that XFXE=AFAE\frac{XF}{XE}=\frac{AF}{AE}. Prove that the circumcircles of triangles AFEAFE and BXCBXC are tangent.
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