57. There are two triangles with respectively parallel sides and areas $S_{1}$ and $S_{2}$, where one of them is inscribed in triangle $A B C$, and the other
is circumscribed around it. Find the area of triangle $A B C$.
57. Let $O$ be the center of similarity of the inscribed and circumscribed triangles, $M_{1}$ and $M_{2}$ be two corresponding vertices ( $M_{1}$ lies on side $A B$ ), and segment $O A$ intersects the inscribed triangle at point $K$. Then $S_{O M_{1} K}=\lambda S_{1}, S_{O M_{2} A}=\lambda S_{2}, \frac{S_{O M_{1} A}}{S_{O M_{2} A}}=\frac{O M_{1}}{O M_{2}}=$ $=\sqrt{\frac{S_{1}}{S_{2}}}$, from which $S_{O M_{1} A}=\lambda \sqrt{S_{1} S_{2}}$, where $\lambda=\frac{S_{O M_{1} K} \text {. }}{S_{1}}$. Considering six such triangles and summing their areas, we get $S_{A B C}=\sqrt{S_{1} S_{2}}$.