Question 12
Let $ABC$ be equilateral, and $D,$ $E,$ and $F$ be the midpoints of $\overline{BC},$ $\overline{CA},$ and $\overline{AB},$ respectively. There exist points $P,$ $Q,$ and $R$ on $\overline{DE},$ $\overline{EF},$ and $\overline{FD},$ respectively, with the property that $P$ is on $\overline{CQ},$ $Q$ is on $\overline{AR},$ and $R$ is on $\overline{BP}.$ The ratio of the area of triangle $ABC$ to the area of triangle $PQR$ is $a + b\sqrt{c},$ where $a,$ $b,$ and $c$ are integers, and $c$ is not divisible by the square of any prime. What is $a^2 + b^2 + c^2?$