2025 AMC 10A Problem 22

Problem 22 of 25HarderGeometry

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 externally tangent to each other, as shown in the diagram below.

What is

r?

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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.