Inequalities For Distances Between Triangle Centers

TL;DR

This paper proves several conjectures about inequalities between distances of triangle centers, using symbolic mathematics, and establishes stronger bounds with optimal constants.

Contribution

It provides rigorous proofs of Kimberling's conjectures and introduces improved inequalities with best-possible constants for distances between triangle centers.

Findings

Proved Kimberling's conjectures on triangle center distances.

Established stronger inequalities with optimal constants.

Validated inequalities using symbolic mathematics techniques.

Abstract

In his seminal paper on triangle centers, Clark Kimberling made a number of conjectures concerning the distances between triangle centers. For example, if $D(i; j)$ denotes the distance between triangle centers $X_i$ and $X_j$ , Kimberling conjectured that $D(6; 1) \leq D(6; 3)$ for all triangles. We use symbolic mathematics techniques to prove these conjectures. In addition, we prove stronger results, using best-possible constants, such as $D(6; 1) \leq (2 -\sqrt3)D(6; 3)$.