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Line passes through fixed point, as point varies

Source: APMO 2022 P2

May 17, 2022
geometrycircumcircleAPMOAPMO 2022

Problem Statement

Let ABCABC be a right triangle with ∠B=90∘\angle B=90^{\circ}. Point DD lies on the line CBCB such that BB is between DD and CC. Let EE be the midpoint of ADAD and let FF be the seconf intersection point of the circumcircle of △ACD\triangle ACD and the circumcircle of △BDE\triangle BDE. Prove that as DD varies, the line EFEF passes through a fixed point.
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