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Circle with diameter AC - AB

Source: 2019 Philippine IMO TST3 Problem 3

May 4, 2022
geometryangle bisectortangentperpendicular

Problem Statement

Given △ABC\triangle ABC with AB<ACAB < AC, let ω\omega be the circle centered at the midpoint MM of BCBC with diameter AC−ABAC - AB. The internal bisector of ∠BAC\angle BAC intersects ω\omega at distinct points XX and YY. Let TT be the point on the plane such that TXTX and TYTY are tangent to ω\omega. Prove that ATAT is perpendicular to BCBC.
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