MathNet 2014 Question 22

Let $ABC$ be a triangle with $AB = AC = \frac{25}{14} BC$. Let $M$ denote the midpoint of $\overline{BC}$ and let $X$ and $Y$ denote the projections of $M$ onto $\overline{AB}$ and $\overline{AC}$, respectively. If the areas of triangle $ABC$ and quadrilateral $AXMY$ are both positive integers, find the minimum possible sum of these areas.



Supplemental illustration; no extra hypothesis.