# Rectangle partitions triangle into four regions

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Rectangle $\mathcal{R}$ is inscribed in triangle $ABC$ such that $\mathcal{R}$ has two vertices on side $BC$ and one on each of the other two sides. Triangle $ABC$ is partitioned into four distinct sections by $\mathcal{R}$. Three of the sections are triangles with areas of $5$, $7$, and $18$ in some order. Compute the largest possible area of $\mathcal{R}$.