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)