perplexus.info :: Geometry : Three Spheres

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?

Submitted by Bractals

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Solution: (Hide)

It is easy to show that |AB| = 2*sqrt(a*b) |BC| = 2*sqrt(b*c) |CA| = 2*sqrt(c*a). Since a < b < c, |AB| < |CA| < |BC|. Therefore, C < B < A. From the cosine rule, A = arccos[(a*b-b*c+c*a)*sqrt(b*c) / (2*a*b*c)]

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
Puzzle Thoughts	K Sengupta	2023-08-06 09:17:17
Curious 180 degree observation	Larry	2005-09-17 14:32:32
re: solution -- Table of values	Charlie	2005-09-15 13:59:23
re: No calculus answer, for the first part	Steve Herman	2005-09-15 03:11:32
No calculus answer, for the first part	Federico Kereki	2005-09-15 02:39:15
solution	Charlie	2005-09-14 18:55:04

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