smc

Let the triangle have side lengths $14,6+x,$ and $8+x$. The area of this triangle can be computed two ways. We have $A=rs$, and $A=\sqrt{s(s-a)(s-b)(s-c)}$, where $s=14+x$ is the semiperimeter. Therefore, $4(14+x)=\sqrt{(14+x)(x)(8)(6)}$. Solving gives $x=7$ as the only valid solution. This triangle has sides $13,14$ and $15$, so the shortest side is $\boxed{13\mathrm{units}}$.