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The Five Touching Spheres (Posted on 2019-08-25) Difficulty: 3 of 5
Three balls of radius 2 are all tangent to one another as well as to a ball of radius 6. A fifth, smaller ball is tangent to all four balls. What is its radius?

No Solution Yet Submitted by Danish Ahmed Khan    
Rating: 5.0000 (1 votes)

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Solution Solution (I'd like confirmation) | Comment 2 of 21 |
I think this is right, but not 100% sure.

Consider the three balls sitting on a plane with the large ball on top.  The fifth ball must be directly underneath the large one.

Call A, B, C the centers of the small ball, the large ball, and any of the three of radius 2 respectively.

The line containing AB must pass through the center of the equilateral triangle containing the centers of the three 2 balls.  Call this point D.  CD=4/sqrt(3)

r is the missing radius.  AB=r+6, AC=r+2, BC=8.  After simplifying, cos<BAC=(r^2+8r+12)/(r^2+8r-12)

<CAD=180-<BAC

sin<CAD=sin<BAC=CD/(r+2)

turning the cosine into the sine of <BAC eventually leads to the quadratic equation 2r^2+15r-9=0 from which r=(-15+3sqrt(33))/4 or about 0.5584



  Posted by Jer on 2019-08-26 19:11:58
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