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Three circles $C_1$, $C_2$ and $C_3$, of radii 1 cm, 2 cm and 3 cm respectively touch as shown. $C_1$ meets $C_2$ at $P$ and meets $C_3$ at $Q$.

What is the length in cm of the longer arc of circle $C_1$ between $P$ and $Q$?

If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.

This problem is taken from the UKMT Mathematical Challenges.