Let $M$,$N$,$Q$ be mid points of segments $BD$,$XY$,$CP$. Since $MN$ is the Gauss line of complete quadrilateral $YCXP.DB$ so $M,N,Q$ are collinear. Since $ABCD$ is a parallelogram so $M$ is also mid point of $CA$.Since $M$ and $Q$ are mid points of $CA$ and $CP$ so $MQ//AP$ but $M,N,Q$ are collinear so $MN//AP$ so the line going through the midpoints of segments $BD$ and $XY$ is either parallel to or coincides with line $AP$.

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