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prove that circucircle is tangent to line

Source: Sharygin Geometry Olympiad 2016 First Round P17 grades 9-11

July 25, 2018
geometrycircumcircletangentcircles

Problem Statement

Let DD be an arbitrary point on side BCBC of triangle ABCABC. Circles ω1\omega_1 and ω2\omega_2 pass through AA and DD in such a way that BABA touches ω1\omega_1 and CACA touches ω2\omega_2. Let BXBX be the second tangent from BB to ω1\omega_1, and CYCY be the second tangent from CC to ω2\omega_2. Prove that the circumcircle of triangle XDYXDY touches BCBC.
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