Publication Date
1-1-2024
Document Type
Article
Publication Title
Journal for Geometry and Graphics
Volume
28
Issue
1
First Page
19
Last Page
27
Abstract
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
centroid, circumcenter, circumradius, circumsphere, ex-center, ex-radius, ex-sphere, in-center, in-radius, in-sphere, isosceles tetrahedron, Lagrange multipliers, orthocenter, orthocentric tetrahedron, regular tetrahedron, twin tetrahedron
Creative Commons License
This work is licensed under a Creative Commons Attribution-Share Alike 4.0 License.
Department
Mathematics and Statistics
Recommended Citation
Hidefumi Katsuura. "In and Ex Spheres of a Tetrahedron" Journal for Geometry and Graphics (2024): 19-27.
