## 2013 AIME I Problem 12

2013 AIME I 12. Let $\Delta PQR$ be a triangle with $\angle P = 75^\circ$ and $\angle Q = 60^\circ$. A regular hexagon $ABCDEF$ with side length 1 is drawn inside $\Delta PQR$ so that side $\overline{AB}$ lies on $\overline{PQ}$, side $\overline{CD}$ lies on $\overline{QR}$, and one of the remaining vertices lies on $\overline{RP}$. There are positive integers $a$, $b$, $c$, and $d$ such that the area of $\Delta PQR$ can be expressed in the form $\frac{a + b\sqrt{c}}{d}$, where $a$ and $d$ are relatively prime and $c$ is not divisible by the square of any prime. Find $a + b + c + d$.