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Nice Geometry

Source: 2014 Korean Mathematical Olympiad #5.

January 22, 2015
geometrygeometry proposed

Problem Statement

There is a convex quadrilateral ABCD ABCD which satisfies ∠A=∠D \angle A=\angle D . Let the midpoints of AB,AD,CD AB, AD, CD be L,M,N L,M,N . Let's say the intersection point of AC,BD AC, BD be E E . Let's say point F F which lies on ME→ \overrightarrow{ME} satisfies ME‾×MF‾=MA‾2 \overline{ME}\times \overline{MF}=\overline{MA}^{2} .
Prove that ∠LFM=∠MFN \angle LFM=\angle MFN . :)
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