Circumcircle-Incircle Analogy for 3-D Cuboid
Source: Tournament of Towns Spring 2015 Senior A-level
February 24, 2017
geometry3-Dimensional Geometrycircumcircle
Problem Statement
It is well-known that if a quadrilateral has the circumcircle and the incircle with the same centre then it is a square. Is the similar statement true in 3 dimensions: namely, if a cuboid is inscribed into a sphere and circumscribed around a sphere and the centres of the spheres coincide, does it imply that the cuboid is a cube? (A cuboid is a polyhedron with 6 quadrilateral faces such that each vertex belongs to edges.)
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