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 xaxis).
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.
3 circles and a tangent

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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. :)
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. :)

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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
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

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