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Beautiful problem with the incircle

Source: Sharygin Correspondence Round 2024 P17

March 6, 2024
geometryincircle

Problem Statement

Let ABCABC be a non-isosceles triangle, ω\omega be its incircle. Let D,E,D, E, and FF be the points at which the incircle of ABCABC touches the sides BC,CA,BC, CA, and ABAB respectively. Let MM be the point on ray EFEF such that EM=ABEM = AB. Let NN be the point on ray FEFE such that FN=ACFN = AC. Let the circumcircles of △BFM\triangle BFM and △CEN\triangle CEN intersect ω\omega again at SS and TT respectively. Prove that BS,CT,BS, CT, and ADAD concur.
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