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## Homework Statement

What is the next radius outwards of this Apollonian gasket?

R = radius of outer circle = 5

r

_{1}= radius of largest inner circle = 3

r

_{2}= radius of second largest inner circle = 1

a = unknown radius

## Homework Equations

C = 2πr

A = πr

^{2}

d = 2r

## The Attempt at a Solution

Make a triangle with 2 sides of length r

_{2}+a and 1 side of length d(d for diameter)

Set the perimeter equal to x:

$$(r_2+a)+(r_2+a)+d = x$$

Plug in known values:

$$(1+a)+(1+a)+2 = x$$

$$5+2a = x$$

$$2a=x-5$$

$$a = \frac{1}{2}x-\frac{5}{2}$$

Substitute into original equation:

$$(r_2+\frac{1}{2}x-\frac{5}{2})+(r_2+\frac{1}{2}x-\frac{5}{2})+d=x$$

Simplify:

$$-3+2 = 0$$

$$-1=0$$

Okay something's wrong. I should be able to solve for x, not get something as ridiculous as -1 = 0 so that I can then solve the perimeter equation for a to find the unknown radius.

So what am I doing wrong here? do I need another linear equation in order to solve this?