1977 AMC 12 Problem 15

Question

Each of the three circles in the adjoining figure is externally tangent to the other two, and each side of the triangle is tangent to two of the circles. If each circle has radius three, then the perimeter of the triangle is

36+9\sqrt2

36+6\sqrt3

36+9\sqrt3

18+18\sqrt3

45

Answer: D

Concepts:tangent circles equilateral triangle incircle, incenter, and inradius

Difficulty rating: 2080

Small hint:

The three circle centers form an equilateral triangle of side

6

Big hint:

View the outer sides as parallel offsets and compare the inradii of two equilateral triangles

Solution:

The circle centers form an equilateral triangle of side

6,

whose inradius is

\frac{6\sqrt3}{6}=\sqrt3.

Each side of the outer triangle is a parallel tangent line three units farther out, so the outer triangle has inradius

3+\sqrt3.

An equilateral triangle of inradius

r

has side

2\sqrt3r,

hence each outer side is

2\sqrt3(3+\sqrt3)=6+6\sqrt3.

The perimeter is

3(6+6\sqrt3)=18+18\sqrt3.

Therefore, the correct answer is D.