Through a certain point taken inside a triangle, three lines parallel to the sides are drawn. These lines divide the triangle into six parts, three of which are triangles with areas $S_{1}, S_{2}, S_{3}$. Find the area $S$

of the given triangle.
Each of the resulting three triangles is similar to the given one.

## Solution

Each of the resulting triangles is similar to the given one. The ratio of the sides of similar triangles is equal to the square root of the ratio of their areas.

Since the sum of the corresponding sides of the resulting triangles is equal to the corresponding side of the original triangle (see fig.), then $\sqrt{S_{1}}+\sqrt{S_{2}}+\sqrt{S_{3}}=\sqrt{S}$.

![](https://cdn.mathpix.com/cropped/2024_05_06_5e9166ba6ad1f1b1f21fg-47.jpg?height=403&width=688&top_left_y=1994&top_left_x=685)

## Answer

$\left(\sqrt{S_{1}}+\sqrt{S_{2}}+\sqrt{S_{3}}\right)^{2}$