Let $\Pi:=P_1P_2\ldots P_5$ be the original pentagon whose vertices have complex coordinates $\varpi_i$, $i=1,2,\ldots,5$. The midpoint $M_i$ of the segment $P_{i-1}P_i$ is then $\mu_i = \frac{\varpi_{i-1}+\varpi_{i}}{2}$, and so the centroid $Z_i$ is $\zeta_i = \frac{\mu_i + \mu_{i+1}+\mu_{i+3}}{3} = \frac{\varpi_i+5\gamma}{6}$, where $G$ is the barycenter of the original pentagon with the complex coordinate $\gamma$. Thus, the barycenter of $Z_1Z_2\ldots Z_5$ is \[\frac{1}{5}\sum_{i=1}^5 \left(\frac{\varpi_i+5\gamma}{6}\right)= \gamma\,.\] A way to reconstruct $\Pi$ is then by first constructing the barycenter of the pentagon $Z_1Z_2\ldots Z_5$, which will coincide with the barycenter $G$ of $\Pi$. Now, the homothety with center at $G$ and scaling factor $6$ will map $Z_i$ to $P_i$.