# 2005 AMC 12A Problems/Problem 16

## Contents

## Problem

Three circles of radius are drawn in the first quadrant of the -plane. The first circle is tangent to both axes, the second is tangent to the first circle and the -axis, and the third is tangent to the first circle and the -axis. A circle of radius is tangent to both axes and to the second and third circles. What is ?

## Solution

### Solution 1

Draw the segment between the center of the third circle and the large circle; this has length . We then draw the radius of the large circle that is perpendicular to the x-axis, and draw the perpendicular from this radius to the center of the third circle. This gives us a right triangle with legs and hypotenuse . The Pythagorean Theorem yields:

Quite obviously , so .

### Solution 2

Don't do this unless really really desperate. But I actually solved this with a ruler (try and see!!). Let and find in terms of . The rest is easy.

Solution by franzliszt