Remember my post about Sangakus?

Some time ago our local math association PRIME organized Soiree Sangaku, where the participants could investigate 30 of these Japanese puzzles. Try to solve these for yourselves!

I call the first one “one rule to Ring them all”. The puzzle is to find (and prove) the relation between the rings. In the second and third one, try find a relation between the radii of the orange circles. Warning: the latter aren’t easy at all… Good luck!

If you like these problems but got stuck at some point,
don’t hesitate to contact me ;)