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 Three Spheres (Posted on 2005-09-14)
Three spheres of radii a, b, and c are tangent to the same plane at points A, B, and C respectively. Each sphere is externally tangent to the other two.
If a < b < c, then which internal angle of triangle ABC is largest and what is its value in terms of a, b, and c?

 See The Solution Submitted by Bractals No Rating

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 Curious 180 degree observation Comment 5 of 5 |
Looking at Charlie's last post, I notice that some values of (a, b, c) give a maximum angle of 180 degrees:
(1, 4, 4)  and (1, 2.25, 9)

I wonder what other values result in a 180 degree angle?

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Looking back at Charlie's first post, and Steve Herman's formula, and setting a=1:
When 1=(bc)/(b+c+2sqrt(bc))  then the angle will be 180 degrees.

This corresponds to the smalled sphere just fitting directly beneath the larger 2 tangent spheres.

Edited on September 17, 2005, 2:47 pm
 Posted by Larry on 2005-09-17 14:32:32

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