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Isosceles right triangle and tangency

Source: Pan-American Girls’ Mathematical Olympiad, P2

October 5, 2021
geometrytangencyPAGMO

Problem Statement

Consider the isosceles right triangle ABCABC with ∠BAC=90∘\angle BAC = 90^\circ. Let ℓ\ell be the line passing through BB and the midpoint of side ACAC. Let Γ\Gamma be the circumference with diameter ABAB. The line ℓ\ell and the circumference Γ\Gamma meet at point PP, different from BB. Show that the circumference passing through A, CA,\ C and PP is tangent to line BCBC at CC.
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