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Another variant of the same figure

Source: 2020 China Southeast 11.2

August 9, 2020
geometryincenter

Problem Statement

In a scalene triangle ΔABC\Delta ABC, AB<ACAB<AC, PBPB and PCPC are tangents of the circumcircle (O)(O) of ΔABC\Delta ABC. A point RR lies on the arc AC^\widehat{AC}(not containing BB), PRPR intersects (O)(O) again at QQ. Suppose II is the incenter of ΔABC\Delta ABC, ID⊥BCID \perp BC at DD, QDQD intersects (O)(O) again at GG. A line passing through II and perpendicular to AIAI intersects AG,ACAG,AC at M,NM,N, respectively. SS is the midpoint of arc AR^\widehat{AR}, andSNSN intersects (O)(O) again at TT. Prove that, if AR∥BCAR \parallel BC, then M,B,TM,B,T are collinear.
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