geo puzzle

In the figure above the circles have centers M, N.
Proof, that the marked areas are equal.

A first approach would be to define the radius of circle M = 1, than calculating the radius of circle N
using the Pythagoras lemma.
But a simplification is possible:

We also may proof, that the marked areas in above pictures are equal.

We do not need the Pythagoras lemma either.
In picture below we observe that area ABCD = 2* area PQRS.

Adding circles:

So, the large red circle has twice the area of the small green circle.
A quarter of the red circle equals half of the green circle.
That concludes the proof.