Inscribed and Circumscribed Spheres of a Tetrahedron

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A tetrahedron is defined by the six edge lengths , , , , , . The inscribed and circumscribed spheres of the tetrahedron are constructed. The incenter is shown as a blue dot, and the circumcenter is a red dot. When do the centers of the inscribed and circumscribed spheres coincide?

Contributed by: Izidor Hafner (April 2017)
Open content licensed under CC BY-NC-SA


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Details

The centers of the inscribed and circumscribed spheres coincide if and only if the tetrahedron is isohedral (when all its faces are the same) or an isosceles tetrahedron [1, p. 103, pp. 118–120].

Reference

[1] V. V. Prasolov and I. F. Sharygin, Problems in Stereometry (in Russian), Moscow: Nauka, 1989.



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