Let $D$ be a point in $ABC$. It is clear that $AB+AC>DB+DC$. Can one give a nontrivial lower bound for $d=AB+AC-DB-DC$ in terms of $AD$ and $BC$?
As pointed out, it seems that $d>\min\{k_1\cdot AD, k_2\cdot\frac{AD^2}{BC}\}$ for some constants $1<k_1, k_2<2$. Not sure if this is true, or there is any sharper estimate.
The initial question suggested that maybe $d>(\overline{AD})^2/\overline{BC}$. But one can easily find counterexamples with $d<(\overline{AD})^2/\overline{BC}$; in fact, one find counterexamples with $d<\epsilon\cdot(\overline{AD})^2/\overline{BC}$ for any arbitratrily small $\epsilon>0$: let $\overline{BC}=1$; $\overline{AD}=x$; $\overline{AB}=\overline{AC}=x^2$. Then, $\lim_{x\to\infty} \frac{d}{(\overline{AD})^2/\overline{BC}}=\lim_{x\to\infty}\frac{2x}{x^2}=0$.