Contact Circles
These four touching circles have another circle hiding amongst them...
Problem
Image
Four touching circles all have radius 1 and their centres are at the corners of a square.
What is the radius of the circle through the points of contact $X, Y, Z$ and $T$?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
Image
The square whose vertices are the centres of the original four circles has side of length 2 units and this distance is equal to the diameter of the circle through $X, Y, Z$ and $T$.
Therefore the radius is 1 unit long.