19. $\tan \frac{A}{2} \tan \frac{B}{2} \tan \frac{C}{2}=\frac{r}{p}$.
Prove that the left side of the equation $=\sqrt{\frac{(p-b)(p-c)}{p(p-a)}} \cdot \sqrt{\frac{(p-a)(p-c)}{p(p-b)}} \cdot \sqrt{\frac{(p-a)(p-b)}{p(p-c)}}$ $=\sqrt{\frac{p(p-a)(p-b)(p-c)}{p^{4}}}=\frac{s}{p^{2}}=\frac{p r}{p^{2}}=\frac{r}{p}=$ the right side of the equation.