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Tri-sphere (Posted on 2024-12-09)

Three spheres S₁, S₂, and S₃ are pairwise externally tangent and are tangent to a plane P at points A, B, and C, respectively. If AB=13, BC=14, and CA=15, compute the product of the radii of the three spheres.

No Solution Yet Submitted by Danish Ahmed Khan

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Solution

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Call the radii p, q, r respectively.

A cross section through S1,S2,B,A reveals a trapezoid with right angles at A and B. A line through S1 parallel to AB make a right triangle who pythagorean equation is 13^2 + (q-p)^2 = (q+p)^2

pq=13^2/4

Two more cross sections similarly give

pr=15^2/4

and

qr=14^2/4

The product of these gives (pqr)^2

and the square root

pqr=(13*14*15)/(2*2*2) = 1365/4

Posted by Jer on 2024-12-09 11:40:46

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