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Interesting Right triangle geometry

Source: EMC 2023 Seniors P2

December 18, 2023
emc2023geometryright trianglecircumcircle

Problem Statement

Let ABCABC be a triangle such that ∠BAC=90∘\angle BAC = 90^{\circ}. The incircle of triangle ABCABC is tangent to the sides BC‾\overline{BC}, CA‾\overline{CA}, AB‾\overline{AB} at D,E,FD,E,F respectively. Let MM be the midpoint of EF‾\overline{EF}. Let PP be the projection of AA onto BCBC and let KK be the intersection of MPMP and ADAD. Prove that the circumcircles of triangles AFEAFE and PDKPDK have equal radius.
Kyprianos-Iason Prodromidis
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