Contact circles
These four touching circles have another circle hiding amongst them...
Age
14 to 16
Challenge level
Problem
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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
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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.