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incenter lies on a segment connecting midpoints (Chile 2010 L2 P3)

Source:

May 31, 2019
geometryincentermidpoints

Problem Statement

The sides BC,CABC, CA, and ABAB of a triangle ABCABC are tangent to a circle at points X,Y,ZX, Y, Z respectively. Show that the center of such a circle is on the line that passes through the midpoints of BCBC and AXAX.
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