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What happened to geometry

Source: 2022 G5

July 9, 2023
geometryIMO ShortlistISL 2022

Problem Statement

Let ABCABC be a triangle and ℓ1,ℓ2\ell_1,\ell_2 be two parallel lines. Let ℓi\ell_i intersects line BC,CA,ABBC,CA,AB at Xi,Yi,ZiX_i,Y_i,Z_i, respectively. Let Δi\Delta_i be the triangle formed by the line passed through XiX_i and perpendicular to BCBC, the line passed through YiY_i and perpendicular to CACA, and the line passed through ZiZ_i and perpendicular to ABAB. Prove that the circumcircles of Δ1\Delta_1 and Δ2\Delta_2 are tangent.
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