 Posers and Puzzles

1. 11 Oct '07 11:081 edit
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.
2. 11 Oct '07 12:25
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!]
3. 11 Oct '07 17:50
Originally posted by TheMaster37
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.
...[text shortened]... problems have a very easy, intuitive solution. The "easy when you see how" kind of problems.
'Line L touches the three circles on the right side'

I'm sorry but I cannot visualise this. What do you mean 'touch'? Tangent? Cut? If tangent then S=T so you dont mean that. ...
4. 11 Oct '07 20:48
Originally posted by wolfgang59
'Line L touches the three circles on the right side'

I'm sorry but I cannot visualise this. What do you mean 'touch'? Tangent? Cut? If tangent then S=T so you dont mean that. ...
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 🙂)
5. 12 Oct '07 06:00
Hmm, I don't have the problem at hand at this time, but I believe the centres of the circles are on one straight line.
6. 12 Oct '07 06:00
Originally posted by wolfgang59
'Line L touches the three circles on the right side'

I'm sorry but I cannot visualise this. What do you mean 'touch'? Tangent? Cut? If tangent then S=T so you dont mean that. ...
By touch I mean tangent 🙂
7. 12 Oct '07 08:32
I read the problem once more.

There are two tangent lines in question, one on the right side and one on the left side of the three circles.

This automatically implies that the centres are colineair.

BTW, I do have the solution now. I'll let you work on it for yourselfes for a while first 🙂
8. 12 Oct '07 08:57
Originally posted by TheMaster37
I read the problem once more.

There are two tangent lines in question, one on the right side and one on the left side of the three circles.

This automatically implies that the centres are colineair.

BTW, I do have the solution now. I'll let you work on it for yourselfes for a while first 🙂
In that case I'll stick with my original solution.
9. 13 Oct '07 10:511 edit
Originally posted by TheMaster37
I read the problem once more.

There are two tangent lines in question, one on the right side and one on the left side of the three circles.

This automatically implies that the centres are colineair.

BTW, I do have the solution now. I'll let you work on it for yourselfes for a while first 🙂
In that case radii of the three circles are sqrt(2/3), sqrt(3/2) and (4/9)*sqrt(2/3) respectively.
10. 13 Oct '07 16:52
Originally posted by ranjan sinha
In that case radii of the three circles are sqrt(2/3), sqrt(3/2) and (4/9)*sqrt(2/3) respectively.
9/4*sqrt(2/3) for the last one (must be bigger, not smaller)

Otherwise, I believe that's the solution I already posted 🙂
11. 13 Oct '07 17:52
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 🙂)
Thanks for that. I have a poor imagination!
12. 14 Oct '07 08:30
Well done!

I knew that once I posted the problem on here it would be a matter of days before the solution came up 🙂