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2 circles concurrent with 1 line wanted, equal angles, isosceles related

Source: 2020 Czech and Slovak Olympiad III A p5

November 24, 2020
geometryconcurrentequal anglesisoscelesconcurrency

Problem Statement

Given an isosceles triangle ABCABC with base BCBC. Inside the side BCBC is given a point DD. Let E,FE, F be respectively points on the sides AB,ACAB, AC that ∣∠BED∣=∣∠DFC∣>90o|\angle BED | = |\angle DF C| > 90^o . Prove that the circles circumscribed around the triangles ABFABF and AECAEC intersect on the line ADAD at a point different from point AA.
(Patrik Bak, Michal Rolínek)
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