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What's that locus?

Source: Sharygin Correspondence Round 2024 P6

March 6, 2024
geometry

Problem Statement

A circle ω\omega and two points A,BA, B of this circle are given. Let CC be an arbitrary point on one of arcs ABAB of ω\omega; CLCL be the bisector of triangle ABCABC; the circle BCLBCL meet ACAC at point EE; and CLCL meet BEBE at point FF. Find the locus of circumcenters of triangles AFCAFC.
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