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Prove that AY is tangent to (AEF)

Source: IMOC 2021 G8

August 11, 2021
geometrytangent

Problem Statement

Let PP be an arbitrary interior point of △ABC\triangle ABC, and APAP, BPBP, CPCP intersect BCBC, CACA, ABAB at DD, EE, FF, respectively. Suppose that MM be the midpoint of BCBC, ⊙(AEF)\odot(AEF) and ⊙(ABC)\odot(ABC) intersect at SS, SDSD intersects ⊙(ABC)\odot(ABC) at XX, and XMXM intersects ⊙(ABC)\odot(ABC) at YY. Show that AYAY is tangent to ⊙(AEF)\odot(AEF).
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