Yes.
Note that equality holds if $c\in [a,b]$.
Furthermore, note that we can assume that arcs $ac$ and $bc$ are tangent to the arc $ab$ at $a$ and $b$ respectively.
On the following diagram, if arcs look tangent, then they are assumed to be tangent.
Observe that $|ab|=|ac|+|cx|+|xb|\geqslant |ac|+|cb|$, where $|{*}{*}|$ denotes the length of the arc.