2023 SSMO Accuracy Round Problems/Problem 10

Problem

Let $\triangle ABC$ be a triangle such $AB = 13$ , $BC = 14$ , $CA = 15$ . Let the incircle of $\triangle ABC$ touch $BC$ at $D$ , $AC$ at $E$ , and $AB$ at $F$ . Let $\ell_A$ be the line through the midpoints of $AE$ and $EF$ . Define $\ell_B$ and $\ell_C$ similarily. Let the area of the star created by the union of $\triangle ABC$ and the triangle bound by $\ell_A$ , $\ell_B$ , and $\ell_C$ be $\frac{p}{q}$ for relatively prime $p$ and $q$ . Find $p + q$ .

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2023 SSMO Accuracy Round Problems/Problem 10

Problem

Solution