3 circles and 3 squares all inscirbed into a right angled triangle problem

Question

This is quite a tricky question for me, but this is how far I got:

My drawing may not be precise, but I do know the points of tangency. I am a little stuck now, and I would appreciate it if someone can guide me through the answer. Thanks.

Answer 1

The figure is self-similar; it contains a smaller version of itself. The ratio of the radius of middle circle to that of the largest circle is the same as ratio of the radius of the smallest circle to that of the middle circle. That is, if we let $r$ be the radius of the middle circle, then $$\frac r{99}=\frac{19}r.$$ Multiplying both sides by $99r$ gives us $$r^2=19\times 99=1881,$$ so $$r=\sqrt{1881}.$$

Answer 2

This construction occurs three times: