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 $$a^{2} + b^{2} + c^{2}$$?

(第十六届AIME1998 第12题)