How would I calculate the area of the shaded region $S\_x$?

I'm trying to use similarity of triangles to find some relationship between the sides, but the equations are getting even more complicated.

In the image it seems that $r\_1$ is equal to $r\_2$, but it is not. And so, with $r\_1 \neq r\_2$, it becomes more difficult.

In Geogebra I saw that the solution holds for $r\_1 \neq r\_2$.

Solution: $3r\_1 r\_2$

[![enter image description here](https://i.stack.imgur.com/SVMDN.png)](https://i.stack.imgur.com/SVMDN.png)