Inequalities For Distances Between Triangle Centers

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 $D(6;1)\le D(6;3)$ for all triangles. We use symbolic mathematics techniques to prove these conjectures. In addition, we prove stronger results, using best-possible constants, such as $D(6;1)\le (2-\sqrt{3})D(6;3)$.