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Homework Statement
What is the next radius outwards of this Apollonian gasket?
R = radius of outer circle = 5
r1 = radius of largest inner circle = 3
r2 = radius of second largest inner circle = 1
a = unknown radius
Homework Equations
C = 2πr
A = πr2
d = 2r
The Attempt at a Solution
Make a triangle with 2 sides of length r2+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?