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tangent circumcircles wanted

Source: 2012 Oral Moscow Geometry Olympiad grades 10-11 p6

September 25, 2019
geometrycircumcircletangent circles

Problem Statement

Tangents drawn to the circumscribed circle of an acute-angled triangle ABCABC at points AA and CC, intersect at point ZZ. Let AA1,CC1AA_1, CC_1 be altitudes. Line A1C1A_1C_1 intersects ZA,ZCZA, ZC at points XX and YY, respectively. Prove that the circumscribed circles of the triangles ABCABC and XYZXYZ are tangent.
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