10.058. A square with side length $m$ is inscribed in an equilateral triangle. Find the side length of the triangle.
Solution.

Given $E F=F N=N M=E M=m$ (Fig. 10.57); $A B=B C=A C=x$. $B O=\sqrt{x^{2}-\frac{x^{2}}{4}}=\frac{x \sqrt{3}}{2} \cdot \Delta B O C \sim \triangle F N C \Rightarrow \frac{x \sqrt{3}}{2 m}=\frac{2 x}{2(x-m)}, x=\frac{m(2 \sqrt{3}+3)}{3}$.

Answer: $\frac{m(2 \sqrt{3}+3)}{3}$.