520. A circle of radius $r$ is inscribed in a triangle with perimeter $p$ and area $S$. How are these three quantities related?
$\triangleright$ By connecting the center of the circle to the vertices of the triangle, we get three triangles, each of which has a height $r$ dropped to one of the sides of the triangle $(a, b, c)$. The areas of these three parts are respectively $a r / 2, b r / 2$, and $cr / 2$. Adding them up, we get $S=(a+b+c) r / 2=p r / 2 . \triangleleft$

Using the concept of area, we can find the ratio of the diagonal of a square to its side. Here's how it is done.