## 3 circles and a tangent

DanishDynamite
Posts: 2608
Joined: Mon Jun 07, 2004 4:58 pm
Location: Copenhagen

### 3 circles and a tangent

Suppose you have 3 unit circles (i.e radius = 1) whose centers are on a line and where each circle just touches at least one other circle. (Imagine 3 pennies lined up along the x-axis).

Draw a line A which passes through the leftmost circle's leftmost point and is tangent to the rightmost circle.

What is the length of the line segment of A contained within the middle circle?

If my calculations are right, the answer is pleasingly rational.
ceptimus
Posts: 1462
Joined: Wed Jun 02, 2004 11:04 pm
Location: UK
I get sqrt(3). Here's how:
http://www.mround.pwp.blueyonder.co.uk/ ... iangle.gif

The big outer triangle gives us sin(a) = 1/4, then using the Sine Rule on the left hand triangle we get:

2 / sin(b) = 1 / sin(a)

So sin(b) is 1/2 and therefore sin(c) is also 1/2, and c = 30 degrees.

Now noting the centre triangle is isosceles, we can see that d = 120 degrees and using the Sine Rule again we get L / sin(120) = 1 / sin(30)

So L = sin(120) / sin(30) = sqrt(3)

Nice problem. :)
Skeptoid
Posts: 1296
Joined: Fri Jun 04, 2004 5:28 am
Location: Wisconsin
ceptimus,

Something doesn't seem right. Where do the 2s come from? Where are the 3 unit circles in relation to your diagram? The way I'm reading the problem, DD claims the solution is rational. Your solution is irrational.
DanishDynamite
Posts: 2608
Joined: Mon Jun 07, 2004 4:58 pm
Location: Copenhagen
Like Skeptoid, I think you've made a mistake ceptimus. The center of the middle circle is 3 units from the leftmost circle's leftmost point.
ceptimus
Posts: 1462
Joined: Wed Jun 02, 2004 11:04 pm
Location: UK
I didn't read the question properly. I solved the case where the line goes through the centre of the left hand circle, and is tangent to the right hand one. Here is my diagram with the circles added.

http://www.mround.pwp.blueyonder.co.uk/ ... angle2.gif

Adapting my solution to the real question :oops: we get:

sin(a) = 1/5, sin(b) = 3/5, L/2 = sqrt(1 - (3/5)^2), so L = 8/5 = 1.6
DanishDynamite
Posts: 2608
Joined: Mon Jun 07, 2004 4:58 pm
Location: Copenhagen
Nicely done, ceptimus. :)

I think your first solution is actually quite pleasing as well.