(3) - Outcoming

Since $AD$ is the bisector of $\triangle ABC$ , $\frac{AB}{AC} = \frac{EB}{EC}$ , which means $\frac{AB + AC}{AC} = \frac{EB + EC}{EC}$ or $EB + EC = \frac{EC}{AC}(AB + AC)$ . But since $EB - EC = 189$ , $\frac{AB}{AC} = \frac{EB}{EC}$ is also $\frac{AB}{AC} = \frac{EC + 189}{EC}$ which rearranges to $\frac{EC}{AC} = \frac{189}{AB - AC}$ . Substituting this back in gives us $EB + EC = \frac{189(AB + AC)}{AB - AC}$ .

Since $AE$ is the altitude of $\triangle ABC$ , $AD^2 + DC^2 = AC^2$ and $AD^2 + DB^2 = AB^2$ . Combining these equations gives $DB^2 - DC^2 = AB^2 - AC^2$ , or $(DB - DC)(DB + DC) = (AB - AC)(AB + AC)$ . Substituting $DB + DC = EB + EC = \frac{189(AB + AC)}{AB - AC}$ from above and the given $DB - DC = 1029$ gives $1029 \cdot \frac{189(AB + AC)}{AB - AC} = (AB - AC)(AB + AC)$ , which simplifies to $194481 = (AB - AC)^2$ , and since $AB > AC$ (otherwise $EB + EC$ would be negative from the equation above), this further simplifies to $AB - AC = \boxed{441}$ .