Journal for Geometry and Graphics 30 (2026), No. 1, 045--059
Copyright by the authors licensed under CC BY SA 4.0
Application of Heron's Formula
Hidefumi Katsuura
San Jose State University, San Jose, U.S.A.
hidefumi.katsuura@sjsu.edu
We apply Heron’s formula to obtain a generalized Ptolemy’s theorem, to prove Euler’s formula for the volume of a tetrahedron, and to prove Klain’s theorem on reversible tetrahedra. After that, we prove triangle inequalities derived from a tetrahedron, find an inequality on the circumradius of a tetrahedron, find a volume expression of an isosceles tetrahedron, and give a condition for four points in the space to be co-planar.
Keywords: Heron’s formula, Euler’s formula, reversible tetrahedron, isosceles tetrahedron, Bang’s theorem, Klain’s theorem, circumradius of a tetrahedron, parallelogram law.
MSC: 51M04.
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