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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.

parallelogram

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    $\begingroup$ "remains to prove that PS and RQ are parallel" $\;-\;$ Try to show that $PQ=RS$ instead. $\endgroup$ Commented Dec 15, 2021 at 21:55
  • $\begingroup$ if you can use coordinate geometry, show that the two slopes are equal $\endgroup$ Commented Dec 15, 2021 at 21:55
  • $\begingroup$ @user29418 thank you for your interest in my question; unfortunately I cannot use coordinate geometry $\endgroup$ Commented Dec 15, 2021 at 22:13
  • $\begingroup$ @dxiv thank you for your interest in my question; to prove that, it seems to me, I should start by noting the figure in the center is a rectangle and then prove that the two triangles to the sides of the rectangle are congruent: the two triangles have and equal side (to side of the rectangle) and an equal angle (the 90° one) but I can't see a third equal element to prove congruence. Is there another way to show this? Thanks $\endgroup$ Commented Dec 15, 2021 at 22:34
  • $\begingroup$ Use first criterium given here: mathwarehouse.com/geometry/quadrilaterals/parallelograms/… $\endgroup$ Commented Dec 15, 2021 at 22:56

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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.

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