HMMT February

Geometry Difficulty 5.3 AIME, harder Find the answer

Let $A B C$ be a triangle with $A B=7, B C=9$ , and $C A=4$ . Let $D$ be the point such that $A B \| C D$ and $C A \| B D$ . Let $R$ be a point within triangle $B C D$ . Lines $\ell$ and $m$ going through $R$ are parallel to $C A$ and $A B$ respectively. Line $\ell$ meets $A B$ and $B C$ at $P$ and $P^{\prime}$ respectively, and $m$ meets $C A$ and $B C$ at $Q$ and $Q^{\prime}$ respectively. If $S$ denotes the largest possible sum of the areas of triangles $B P P^{\prime}, R P^{\prime} Q^{\prime}$ , and $C Q Q^{\prime}$ , determine the value of $S^{2}$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Let $R^{\prime}$ denote the intersection of the lines through $Q^{\prime}$ and $P^{\prime}$ parallel to $\ell$ and $m$ respectively. Then $\left[R P^{\prime} Q^{\prime}\right]=\left[R^{\prime} P^{\prime} Q^{\prime}\right]$ . Triangles $B P P^{\prime}, R^{\prime} P^{\prime} Q^{\prime}$ , and $C Q Q^{\prime}$ lie in $A B C$ without overlap, so that on the one hand, $S \leq A B C$ . On the other, this bound is realizable by taking $R$ to be a vertex of triangle $B C D$ . We compute the square of the area of $A B C$ to be $10 \cdot(10-9) \cdot(10-7) \cdot(10-4)=$ 180.

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