HMMT 2021 U9

Let $A D, B E$, and $C F$ be segments sharing a common midpoint, with $A B<A E$ and $B C<B F$. Suppose that each pair of segments forms a $60^{\circ}$ angle, and that $A D=7, B E=10$, and $C F=18$. Let $K$ denote the sum of the areas of the six triangles $\triangle A B C, \triangle B C D, \triangle C D E, \triangle D E F, \triangle E F A$, and $\triangle F A B$. Compute $K \sqrt{3}$.