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Journal for Geometry and Graphics 22 (2018), No. 2, 219--227
Copyright Heldermann Verlag 2018

Theorems on Two Tetrahedrons Intersecting a Sphere

Ken Morita
Graduate School of Electrical and Electronic Engineering, Chiba University, Chiba 263-8522, Japan

We describe three-dimensional theorems of two tetrahedrons intersecting a sphere. These theorems can be considered as generalizations of the two-dimensional Pascal's hexagon and Steiner's theorems. We first restructure the original version of the two-dimensional Pascal's hexagon theorem, and prove it synthetically using a simple lemma. In the proving process, we found the essential nature of Pascal's theorem that leads to the synthetic generalization in three-dimensional space. In order to focus on visual representations, we only use a synthetic method in the generalization process.

Keywords: Triangles and tetrahedrons in perspective, extension of Pascal's hexagon and Steiner's theorems.

MSC: 51M04; 51M35

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