Intercenter Geometry

TL;DR

The paper introduces Intercenter Geometry, a new geometric framework expressing plane and space quantities via triangle sides and tetrahedron edges, solving longstanding problems and offering theoretical and practical advancements.

Contribution

It presents a novel geometry that unifies the calculation of geometric quantities using side lengths, solving long-standing problems in Euclidean and analytical geometry.

Findings

New theorems and formulas on geometry and inequalities

Solved problems like the distance between centroid and incenter of a tetrahedron

Enriched the field of geometry with theoretical and practical significance

Abstract

The author proposes a new geometry in this book. The author named this new geometry Intercenter Geometry. Intercenter Geometry is different from traditional Euclidean geometry and analytic geometry (coordinate geometry). The idea of Intercenter Geometry is that the geometric quantities on a plane will be expressed by the lengths of the three sides of a given triangle. The geometric quantities in space will be expressed by the lengths of the six edges of a given tetrahedron. In Intercenter Geometry, a unified approach is used to deal with the calculation of geometric quantities of triangles and tetrahedrons. Many new theorems and formulas on geometry and geometric inequalities have been obtained. In particular, Intercenter Geometry has solved some problems that have not been solved in Euclidean geometry and analytical geometry (coordinate geometry) for a long time, such as the distance between the centroid and the incenter of a given tetrahedron, etc. Intercenter Geometry enriches geometry, which is not only of theoretical significance to open up the field of mathematical research, to explore new ideas and methods of mathematical research, but also of positive significance to promote the spirit of innovation. The fruitful achievements of Intercenter Geometry also have practical application value.