I didn't proceed that way, but I imagine this may work too. I'll post the answer tomorrow.

south_korea_ln

science

[Not-so-daily puzzle] A fun geometry puzzle

It might help to assemble four quarter circles into a full one:

![circle](https://i.ibb.co/mzBgwKd/circle.png)

Let's say the doubly shaded right triangles have shorter sides $$x \gt y$$.

The outer square's side length is $$2x+12$$, area $$4x^2+48x+144$$.

We have to subtract the four white rectangles with area $$1+x$$ each.

Further add the doubly shaded triangle areas $$2xy$$ (they count twice).

Then the answer is obviously a fourth of that.

That's how far I got, but I couldn't get $$x$$ and $$y$$.