I've been at this for a while now but I cannot find an answer as of yet.

Three different circles, A, B, C in order of increasing radii are against eachother. A and B touch in point S, B and C touch in point T.

Line L touches the three circles on the right side. It touches A in point X, it touches B in point Y and C in point Z.

XY = 2, YZ = 3.

Find the radius of the smallest circle.

EDIT: This is a problem from a set (no, not my homework). Alot of the problems have a very easy, intuitive solution. The "easy when you see how" kind of problems.

You know, unless I'm missing a bit of information then I don't think there's a unique answer.

I can get two relationships out of this. If a, b and c are the radii of the three circles, then:

ab = 1, bc = 9/4.

a < b < c implies that 2/3 < a < 1, but it seems like you can choose any a in this range and make it work.

E.g. a = 4/5, b = 5/4, c = 9/5.

Plotting a couple of solutions of this seems to confirm that it works. Unless there's more information, for instance the centres of the circles being in a straight line. This would lead to the specific solution:

a = sqrt(2/3), b = sqrt(3/2), c = sqrt(27/8)

[I'll post the working later - these geometry questions are awkward to explain without diagrams!]

Originally posted by mtthw I assumed tangent - it doesn't mean S = T. Think about a horizontal surface with three cylinders sitting on it in contact with each other.

(If that's not right you can completely ignore my previous post ðŸ™‚)