For 28 quadrilateral shapes and 1000 triangle centers, the paper catalogs relationships such as area ratios, similarity, concurrence, and homothety between a quadrilateral and its central quadrilateral.
Arrangement of Central Points on the Faces of a Tetrahedron
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abstract
We systematically investigate properties of various triangle centers (such as orthocenter or incenter) located on the four faces of a tetrahedron. For each of six types of tetrahedra, we examine over 100 centers located on the four faces of the tetrahedron. Using a computer, we determine when any of 16 conditions occur (such as the four centers being coplanar). A typical result is: The lines from each vertex of a circumscriptible tetrahedron to the Gergonne points of the opposite face are concurrent.
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More Relationships between a Central Quadrilateral and its Reference Quadrilateral
For 28 quadrilateral shapes and 1000 triangle centers, the paper catalogs relationships such as area ratios, similarity, concurrence, and homothety between a quadrilateral and its central quadrilateral.