  We perform a theoretical study on direct $CP$ violation in $D^\pm\to \pi^\pm
\pi^+\pi^-$ in phase space around the intermediate states $\rho^0(770)$ and
$f_0(500)$. The possible interference between the amplitudes corresponding to
the two resonances is taken into account, and the relative strong phase of the
two amplitudes is treated as a free parameter. Our analysis shows that by
properly chosen the strong phase, both the $CP$ violation strength and
differential decay width accommodate to the experimental results.
