Question

A circle of radius $r$ is surrounded by three circles, whose radii are $1, 2,$ and $3,$ all externally tangent to the inner circle and to each other, as shown.

What is $r?$

$\dfrac{1}{4}$

$\dfrac{6}{23}$

$\dfrac{3}{11}$

$\dfrac{5}{17}$

$\dfrac{3}{10}$

Solution:

The three outer centers $A, B, C$ are pairwise $AB = 1 + 2 = 3,$ $AC = 1 + 3 = 4,$ and $BC = 2 + 3 = 5$ apart, a $3$ - $4$ - $5$ right triangle. Now apply Descartes' Circle Theorem with curvatures $1, \tfrac12, \tfrac13,$ and $\tfrac1r,$ all mutually tangent: $\frac1r = 1 + \tfrac12 + \tfrac13 + 2\sqrt{\tfrac12 + \tfrac16 + \tfrac13} = \tfrac{11}{6} + 2\sqrt{1} = \tfrac{23}{6}.$ Inverting, $r = \tfrac{6}{23}.$ Therefore, the answer is B.

2025 AMC 10A Problem 22

Solution:

Problem 22 in Other Years