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Circles in Circles

Age 16 to 18 Challenge Level:

Take three unit circles, each touching the other two. Construct three circles $C_1$, $C_2$ and $C_3$, with radii $r_1$, $r_2$ and $r_3$, respectively, as in the figure below. The circles that are tangent to all three unit circles are $C_1$ and $C_3$, with $C_1$ the smaller of these. The circle through the three points of tangency of the unit circles is $C_2$. Find the radii $r_1$, $r_2$ and $r_3$, and show that $r_1r_3=r_2^2$.

Circles in a circle