  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 $D(6; 1) \leq (2
-\sqrt3)D(6; 3)$.
