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2 concentric spheres, triangle ABC of big tangent to small

Source: Nordic Mathematical Contest 1988 #3

October 5, 2017

geometry3D geometrysphere

Problem Statement

Two concentric spheres have radii

r

and

R,r < R

. We try to select points

A, B

and

C

on the surface of the larger sphere such that all sides of the triangle

ABC

would be tangent to the surface of the smaller sphere. Show that the points can be selected if and only if

R \le 2r