  We report the first amplitude analysis of the decays $D^{0} \to \pi^{+}
\pi^{-} \eta$ and $D^{+} \rightarrow \pi^{+}\pi^{0}\eta$ using a data sample
taken with the BESIII detector at the center-of-mass energy of 3.773 GeV,
corresponding to an integrated luminosity of 7.9 ${\rm fb}^{-1}$. The
contribution from the process $D^{0(+)} \to a_{0}(980)^{+} \pi^{-(0)}$ is
significantly larger than the $D^{0(+)} \to a_{0}(980)^{-(0)} \pi^{+}$
contribution. The ratios $\mathcal{B}(D^{0} \rightarrow
a_{0}(980)^{+}\pi^{-})/\mathcal{B}(D^{0} \rightarrow a_{0}(980)^{-}\pi^{+})$
and $\mathcal{B}(D^{+} \rightarrow a_{0}(980)^{+}\pi^{0})/\mathcal{B}(D^{+}
\rightarrow a_{0}(980)^{0}\pi^{+})$ are measured to be
$7.5^{+2.5}_{-0.8\,\mathrm{stat.}}\pm1.7_{\mathrm{syst.}}$ and
$2.6\pm0.6_{\mathrm{stat.}}\pm0.3_{\mathrm{syst.}}$, respectively. The measured
$D^{0}$ ratio disagrees with the theoretical predictions by orders of
magnitudes, thus implying a substantial contribution from final-state
interactions.
