Contact Circles
These four touching circles have another circle hiding amongst them...
Image
![Contact Circles Contact Circles](/sites/default/files/styles/large/public/thumbnails/content-id-6789-Contact%252520Circles%252520Stefania.png?itok=HJ-KZl-e)
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.
Image
![Contact Circles Contact Circles](/sites/default/files/styles/large/public/thumbnails/content-id-6789-Contact%252520Circles%252520Solution%252520Stefania.png?itok=XLa8cH3D)
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.