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Perpendicularity in incircle/circumcircle/arc midpt diagram

Source: XVII Olimpíada Matemática Rioplatense (2008)

July 25, 2011
geometrycircumcircletrigonometrycyclic quadrilateralangle bisectorgeometry unsolved

Problem Statement

In triangle ABCABC, where AB<ACAB<AC, let XX, YY, ZZ denote the points where the incircle is tangent to BCBC, CACA, ABAB, respectively. On the circumcircle of ABCABC, let UU denote the midpoint of the arc BCBC that contains the point AA. The line UXUX meets the circumcircle again at the point KK. Let TT denote the point of intersection of AKAK and YZYZ. Prove that XTXT is perpendicular to YZYZ.
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