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Incenter geo, prove that AR is parallel to BC

Source: IMOC 2021 G3

August 11, 2021
geometryincenter

Problem Statement

Let II be the incenter of the acute triangle △ABC\triangle ABC, and BIBI, CICI intersect the altitude of △ABC\triangle ABC through AA at UU, VV, respectively. The circle with AIAI as a diameter intersects ⊙(ABC)\odot(ABC) again at TT, and ⊙(TUV)\odot(TUV) intersects the segment BCBC and ⊙(ABC)\odot(ABC) at PP, QQ, respectively. Let RR be another intersection of PQPQ and ⊙(ABC)\odot(ABC). Show that AR∥BCAR\parallel BC.
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