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Show that (DEN) passes through the midpoint of BC

Source: Sharygin First Round 2013, Problem 21

April 7, 2013
geometrySharygin Geometry Olympiadgeometry solvedpower of a pointreflectionAngle Chasingsimilar triangles

Problem Statement

Chords BCBC and DEDE of circle ω\omega meet at point AA. The line through DD parallel to BCBC meets ω\omega again at FF, and FAFA meets ω\omega again at TT. Let M=ET∩BCM = ET \cap BC and let NN be the reflection of AA over MM. Show that (DEN)(DEN) passes through the midpoint of BCBC.
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