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Rays passing through a point

Source: Saint Petersburg MO 2020 Grade 10 Problem 5

May 7, 2020
geometrycircumcircle

Problem Statement

Rays ℓ,ℓ1,ℓ2\ell, \ell_1, \ell_2 have the same starting point OO, such that the angle between ℓ\ell and ℓ2\ell_2 is acute and the ray ℓ1\ell_1 lies inside this angle. The ray ℓ\ell contains a fixed point of FF and an arbitrary point LL. Circles passing through FF and LL and tangent to ℓ1\ell_1 at L1L_1, and passing through FF and LL and tangent to ℓ2\ell_2 at L2L_2. Prove that the circumcircle of △FL1L2\triangle FL_1L_2 passes through a fixed point other than FF independent on LL.
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