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BC is tangent to circumcircle of DEF iff D is midpoint of BC, AB=AC, <FDE=<ABC

Source: 2013 Grand Duchy of Lithuania, Mathematical Contest p2 (Baltic Way TST)

October 2, 2020
geometryisoscelestangentequal angles

Problem Statement

Let ABCABC be an isosceles triangle with AB=ACAB = AC. The points D,ED, E and FF are taken on the sides BC,CABC, CA and ABAB, respectively, so that ∠FDE=∠ABC\angle F DE = \angle ABC and FEFE is not parallel to BCBC. Prove that the line BCBC is tangent to the circumcircle of △DEF\vartriangle DEF if and only if DD is the midpoint of the side BCBC.
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