If circular arcs $AC$ and $BC$ have centers at $B$ and $A$, respectively, then there exists a circle tangent to both $\stackrel{⌢}{\mathrm{AC}}$ and $\stackrel{⌢}{\mathrm{BC}}$, and to $\overline{AB}$. If the circumference of this tangent circle is $36$, what is the length of $\stackrel{⌢}{\mathrm{BC}}$?