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Let $M$, $N$, and $P$ be the midpoints of sides $BC$, $AC$, and $AB$ of triangle $ABC$, respectively. $E$ and $F$ are two points on the segment $BC$ so that $\angle NEC =\frac{1}{2}\angle AMB$ and $\angle PFB = \frac{1}{2}\angle AMC$. Prove that $AE = AF$.