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Journal for Geometry and Graphics 17 (2013), No. 1, 001--005
Copyright Heldermann Verlag 2013

Generalization of a Geometry Problem Posed by Fermat

Zvonko Cerin
Kopernikova 7, 10010 Zagreb, Croatia

We consider a configuration generated by a point on a sphere and a rectangular hexahedron whose top face is inscribed to the equator of the sphere. Our results could be regarded as the space version of a geometric problem in the plane posed by Fermat in 1658 that was first proved by Euler almost one century later in 1750. In fact, both the ratio of the semicircle's diameter and the rectangle's height in Fermat's theorem and of the sphere's diameter and the hexahedron's height in our generalization of it is equal to the square root of 2. We also give four additional invariant and equivalent properties for this configuration. Moreover, when the above ratio is equal to a third of the square root of 2 another sum of squares does not depend on the position of a point on the sphere.

Keywords: 3-dimensional Euclidean space, sphere, hexahedron, sum of squares of distances, Fermat geometric problem.

MSC: 51M04

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