Consider a parallelogram $ABCD$ and a plane, $P$. Let $A'B'C'D'$ be the orthogonal projective image of $ABCD$ onto $P$. If $P$ cuts the segments $AC$ and $CD$, then $BB'=AA'+CC'+DD'$. If $P$ cuts the segments $AB$ and $CD$, then $CC'+ DD'=AA'+BB'$. I have an idea of a proof using the distance formula from a point to a plane, so my questions are: 1) is this known? 2) Does it generalizes further?
Edit: I have constructed a regular parallelogram in the image, so I do not know if this works for a non-regular.