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Half-equilateral triangles are constructed [ILL 1974]

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January 3, 2011
geometry proposedgeometry

Problem Statement

Outside an arbitrary triangle ABCABC, triangles ADBADB and BCEBCE are constructed such that ∠ADB=∠BEC=90∘\angle ADB=\angle BEC=90^{\circ} and ∠DAB=∠EBC=30∘\angle DAB=\angle EBC=30^{\circ}. On the segment ACAC the point FF with AF=3FCAF=3FC is chosen. Prove that ∠DFE=90∘\angle DFE=90^{\circ} and ∠FDE=30∘\angle FDE=30^{\circ}.
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