Journal for Geometry and Graphics 28 (2024), No. 1, 019--027
Copyright by the authors licensed under CC BY SA 4.0
In and Ex Spheres of a Tetrahedron
Hidefumi Katsuura
San Jose State University, San Jose, U.S.A., San Jose, U.S.A.
hidefumi.katsuura@sjsu.edu
We prove that
(1) a tetrahedron is isosceles if and only if the vertices of its twin tetrahedron are the excenters of the tetrahedron,
(2) if a tetrahedron is orthocentric, and if the orthocenter is either the incenter, the centroid, or the circumcenter, then the tetrahedron is regular,
(3) a tetrahedron is regular if and only if the four ex-spheres are tangent to the in-sphere, and
(4) we prove an inequality relating the in-radius, circumradius, and the distances between the in-center and the vertices of a tetrahedron.
Keywords: In-sphere, in-center, in-radius, ex-sphere, ex-center, ex-radius, twin tetrahedron, isosceles tetrahedron, regular tetrahedron, centroid, circumsphere, circumradius, circumcenter, orthocentric tetrahedron, orthocenter, Lagrange multipliers.
MSC: 51M04.
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