In acute triangle $ABC$, points $D$, $E$, and $F$ are located on sides $\overline{BC}$, $\overline{AC}$, and $\overline{AB}$ so
that $\overline{AD} \perp \overline{BC}$, $\overline{DE} \perp \overline{AC}$, and $\overline{DF} \perp \overline{AB}$. Let $R_1$ and $R_2$ be the radii of the circles around $\triangle ABC$ and $\triangle AEF$, respectively. Determine the number of degrees in the measure of $\angle A$ if the area of $\triangle ABC$ is equal to $R_1R_2$.
