Question 76
Let $ABC$ be a triangle with $AB = 7$, $BC = 9$, and $CA = 4$. Let $D$ be the point such that $AB \parallel CD$ and $CA \parallel BD$. Let $R$ be a point within triangle $BCD$. Lines $\ell$ and $m$ going through $R$ are parallel to $CA$ and $AB$ respectively. Line $\ell$ meets $AB$ and $BC$ at $P$ and $P'$, respectively, and $m$ meets $CA$ and $BC$ at $Q$ and $Q'$, respectively. If $S$ denotes the largest possible sum of the areas of triangles $BPP'$, $RP'Q'$, and $CQQ'$, determine the value of $S^2$.
