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P is incenter of the CDE wanted, starting with a right triangle

Source: Czech-Polish-Slovak Junior Match 2015, Individual p4 CPSJ

March 15, 2020
geometryangle bisectorincenterright trianglemidpoints

Problem Statement

Let ABCABC ne a right triangle with ∠ACB=90o\angle ACB=90^o. Let E,FE, F be respecitvely the midpoints of the BC,ACBC, AC and CDCD be it's altitude. Next, let PP be the intersection of the internal angle bisector from AA and the line EFEF. Prove that PP is the center of the circle inscribed in the triangle CDECDE .
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