The segments EF and PD meet at their midpoint at point M. Prove that PE is parallel to DF.

If the segments and EF and PD intersect at point M, and it is the middle of each of them, then FM = EM and PM = MD. Angle EMD = Angle PMD (they are vertical).
This means that the triangles are equal in the first attribute.
Accordingly, the EMP triangle is also equal to the FMD triangle in the same components.
They have the same cross-lying angles FED and EFP. This means that the lines are parallel.