Intercenter Geometry

Daiyuan Zhang

Home/Archive/arXiv

arXiv · 2107.08388

Intercenter Geometry

Daiyuan Zhang

View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

math.GM

Keep this discovery

BibTeX RIS

Daiyuan Zhang. 2021-07-18. Intercenter Geometry. https://arxiv.org/abs/2107.08388

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Average Chord Lengths in a Triangle

Let $P$ be a point inside a triangle $T$. We consider the average length of the chords of $T$ through $P$, where the direction of the chord is chosen uniformly. An elementary formula is obtained in terms of the distances from $P$ to the sides and vertices of the triangle. Several classical triangle centers give especially simple specializations. For example, if $I$ is the incenter, then \[ M_T(I)=\frac{2r}{\pi} \log\left(\cot\frac A4\cot\frac B4\cot\frac C4\right). \] Our main result is the sharp inequality \[ M_T(P)\le \frac{p}{\pi\sqrt3}\log(2+\sqrt3), \] valid simultaneously for every triangle of perimeter $p$ and every interior point $P$. Thus, among all such pairs $(T,P)$, the largest possible average chord length occurs only when $T$ is equilateral and $P$ is its center. The proof is an elementary symmetrization argument. We close with brief remarks relating the problem to the radial center of a convex body, the electrostatic potential center of a triangle, and dual quermassintegrals.

math.GM↗

arXiv2026

The decrease in sectional curvature of the Fisher--Rao manifold of beta distributions

In the paper, the author develops a general convexity lemma and applies it to demonstrate the monotonic decrease of a function involving polygamma functions, arising from the sectional curvature of the Fisher--Rao manifold of beta distributions.

math.GM↗

arXiv2026

A Proof of Liu's Conjecture on the Fundamental Triangle Inequality

Let $a,b,c$ be the side lengths of a triangle, and let $R$ and $r$ denote its circumradius and inradius, respectively. We prove a conjecture of Liu stating that \[\sum_{\mathrm{cyc}} \left(\frac{a(b+c-a)}{bc}\right)^k \geq 2+\left(\frac{2r}{R}\right)^k,~~k>1, \] with the reverse inequality for $0<k<1$. The proof reduces the problem to three positive variables with fixed sum and product. We also determine the equality cases.

math.GM↗