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Geo with angle and length conditions

Source: Polish MO Second round 2024 P2

February 9, 2024
geometry

Problem Statement

Let ABCDABCD be a convex quadrilateral with ∠ABC=∠ADC=120∘\angle ABC=\angle ADC=120^{\circ}. The point EE lies on the segment ADAD and is such that AE⋅BC=AB⋅DEAE \cdot BC=AB \cdot DE and similarly the point FF lies on the segment BCBC and satisfies BF⋅CD=AD⋅CFBF \cdot CD=AD \cdot CF. Show that BEBE and DFDF are parallel.
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