I'm trying to prove the following statement:
"Given a parallelogram ABCD, through the midpoint of AD draw a perpendicular adn denote with Q its intersection with line AB. Similarly, draw through the midpoint R of BC a line perpendicular to BC and denote with S its intersection with line CD. Show that the quadrilateral PQRS is a parallelogram"
Now, since ABCD is a parallelogram by hypothesis, we know that $AD||BC$ and since $PQ$ and $RS$ are perpendicular to two parallel lines, they are themselves parallel.
Now, it remains to prove that $PS$ and $RQ$ are parallel but I haven't been able to do so, so I would appreciate an hint about how to show this, thanks.
Best Answer
Let $M$ be the common midpoint of $\overline{AC}$ and $\overline{BD}$. Then $M$ is also the midpoint of $\overline{PR}$.
Reflect everything through $M$. Lines $AB$ and $CD$ are swapped. Lines $AD$ and $BC$ are swapped, so lines $PQ$ and $RS$ are swapped. So points $Q$ and $S$ are swapped.
So $M$ is the common midpoint of $\overline{PR}$ and $\overline{QS}$, which implies $PQRS$ is a parallelogram.