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Reflections of incircle tangency points with respect to exte

Source: 3rd QEDMO 2006, problem 6, by me

April 14, 2006
geometrygeometric transformationreflectiontrapezoidhomothetygeometry proposed

Problem Statement

The incircle of a triangle ABCABC touches its sides BCBC, CACA, ABAB at the points XX, YY, ZZ, respectively. Let X′X^{\prime}, Y′Y^{\prime}, Z′Z^{\prime} be the reflections of these points XX, YY, ZZ in the external angle bisectors of the angles CABCAB, ABCABC, BCABCA, respectively. Show that Y′Z′∥BCY^{\prime}Z^{\prime}\parallel BC, Z′X′∥CAZ^{\prime}X^{\prime}\parallel CA and X′Y′∥ABX^{\prime}Y^{\prime}\parallel AB.
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