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Prove collinear

Source: 2023 China South-east MO Grade 10 P3

August 1, 2023
geometryincenter

Problem Statement

In acute triangle ABCABC (△ABC\triangle ABC is not an isosceles triangle), II is its incentre, and circle ω \omega is its inscribed circle. ⊙ω\odot\omega touches BC,CA,ABBC, CA, AB at D,E,FD, E, F respectively. ADAD intersects with ⊙ω\odot\omega at JJ (J≠DJ\neq D), and the circumcircle of △BCJ\triangle BCJ intersects ⊙ω\odot\omega at KK (K≠JK\neq J). The circumcircle of △BFK\triangle BFK and △CEK\triangle CEK meet at LL (L≠KL\neq K). Let MM be the midpoint of the major arc BACBAC. Prove that M,I,LM, I, L are collinear.
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