Question 47
Let $AD$, $BE$, and $CF$ be segments sharing a common midpoint, with $AB \lt AE$ and $BC \lt BF$. Suppose that each pair of segments forms a $60^{\circ}$ angle, and that $AD = 7$, $BE = 10$, and $CF = 18$. Let $K$ denote the sum of the areas of the six triangles $\triangle ABC$, $\triangle BCD$, $\triangle CDE$, $\triangle DEF$, $\triangle EFA$, and $\triangle FAB$. Compute $K \sqrt{3}$.