smc

Drawing $\overline{PA}$, $\overline{PB}$, and $\overline{PC}$, $△ABC$ is split into three smaller triangles. The altitudes of these triangles are given in the problem as $PQ$, $PR$, and $PS$. Summing the areas of each of these triangles and equating it to the area of the entire triangle, we get: $$\frac{s}{2}+\frac{2s}{2}+\frac{3s}{2}=\frac{s^2\sqrt{3}}{4}$$ where $s$ is the length of a side of the equilateral triangle $$s=\boxed{4\sqrt{3}}$$ * Note - This is called Viviani's Theorem.