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Equal lengths in incircle configuration

Source: China second round 2018 (A) Q2

May 5, 2019
geometryincentercircumcirclecongruent triangles

Problem Statement

In triangle △ABC\triangle ABC, AB<ACAB<AC, M,D,EM,D,E are the midpoints of BCBC, the arcs BACBAC and BCBC of the circumcircle of △ABC\triangle ABC respectively. The incircle of △ABC\triangle ABC touches ABAB at FF, AEAE meets BCBC at GG, and the perpendicular to ABAB at BB meets segment EFEF at NN. If BN=EMBN=EM, prove that DFDF is perpendicular to FGFG.
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