A small circle fits between two touching circles so that all three
circles touch each other and have a common tangent? What is the
exact radius of the smallest circle?

Ten squares form regular rings either with adjacent or opposite
vertices touching. Calculate the inner and outer radii of the rings
that surround the squares.

Three semi-circles have a common diameter, each touches the other
two and two lie inside the biggest one. What is the radius of the
circle that touches all three semi-circles?

Kissing

Age 16 to 18 Challenge Level

The diagram is

Without loss of generality you can takethe radii of the two identical circles to be 1 unit.

Using Pythagoras theorem can you then find the radius of the small circle? Don't worry if you get surds in the answer.