Let $ABC$ be [equilateral](https://artofproblemsolving.com/wiki/index.php/Equilateral_triangle), and $D, E,$ and $F$ be the [midpoints](https://artofproblemsolving.com/wiki/index.php/Midpoint) of $\overline{BC}, \overline{CA},$ and $\overline{AB},$ respectively.  There exist [points](https://artofproblemsolving.com/wiki/index.php/Point) $P, Q,$ 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](https://artofproblemsolving.com/wiki/index.php/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](https://artofproblemsolving.com/wiki/index.php/Prime).  What is $a^{2} + b^{2} + c^{2}$?
[1998 AIME-12.png](https://artofproblemsolving.com/wiki/index.php/File:1998_AIME-12.png)
We let $x = EP = FQ$, $y = EQ$, $k = PQ$. Since $AE = \frac {1}{2}AB$ and $AD = \frac {1}{2}AC$, $\triangle AED \sim \triangle ABC$ and $ED \parallel BC$.
By alternate interior angles, we have $\angle PEQ = \angle BFQ$ and $\angle EPQ = \angle FBQ$. By vertical angles, $\angle EQP = \angle FQB$. 
Thus $\triangle EQP \sim \triangle FQB$, so $\frac {EP}{EQ} = \frac {FB}{FQ}\Longrightarrow\frac {x}{y} = \frac {1}{x}\Longrightarrow x^{2} = y$.
Since $\triangle EDF$ is equilateral, $EQ + FQ = EF = BF = 1\Longrightarrow x + y = 1$. Solving for $x$ and $y$ using $x^{2} = y$ and $x + y = 1$ gives $x = \frac {\sqrt {5} - 1}{2}$ and $y = \frac {3 - \sqrt {5}}{2}$.
Using the [Law of Cosines](https://artofproblemsolving.com/wiki/index.php/Law_of_Cosines), we get

$k^{2}  =  x^{2} + y^{2} - 2xy\cos{\frac {\pi}{3}}$
$=  \left(\frac {\sqrt {5} - 1}{2}\right)^{2} + \left(\frac {3 - \sqrt {5}}{2}\right)^{2} - 2\left(\frac {\sqrt {5} - 1}{2}\right)\left(\frac {3 - \sqrt {5}}{2}\right)\cos{\frac {\pi}{3}}$
$= 7 - 3\sqrt {5}$
We want the ratio of the squares of the sides, so $\frac {(2)^{2}}{k^{2}} = \frac {4}{7 - 3\sqrt {5}} = 7 + 3\sqrt {5}$ so $a^{2} + b^{2} + c^{2} = 7^{2} + 3^{2} + 5^{2} = \boxed{083}$.