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Three lines forming an isosceles triangle

Source: IRN-SGP-TWN 2023 friendly contest p5

July 16, 2023
geometry

Problem Statement

I,ΩI,\Omega are the incenter and the circumcircle of triangle ABCABC, respectively, and the tangents of B,CB,C to Ω\Omega intersect at LL. Assume that P≠CP\neq C is a point on Ω\Omega such that CI,APCI,AP, and the circle with center LL and radius LCLC are concurrent. Let the foot from II to ABAB be FF, the midpoint of BCBC be MM, XX is a point on Ω\Omega s.t. AI,BC,PXAI,BC,PX are concurrent. Prove that the lines AI,AX,MFAI,AX,MF form an isosceles triangle.
Proposed by ckliao914
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