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Omni-MATH / It is well-known that if a quadrilateral has the circumcircle and the incircle…

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problem

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 33 edges.) [i](1010 points)[/i]
Plain-text mathematical notation (without MathML)
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 3 edges.)
[i](10 points)[/i]
Original LaTeX notation
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 $3$ edges.)
[i]($10$ points)[/i]

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Source and history

task_id
b8129d81-8f62-58cb-9b28-013241f22a8d
task_key
test--b8129d81-8f62-58cb-9b28-013241f22a8d
task_revision_id
1
config / split
/ test

Official source

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