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It's well known that consecutively connecting midpoints of an arbitrary quadrilateral forms a parallelogram. Is it possible to inscribe other parallelograms inside a quadrilateral? I didn't find an example for that, so tried to come up with some argument which shows that parallelogram is unique but didn't get result. After that I used complex numbers to describe the problem. Let $ABCD$ be a quadrilateral with points $M$, $P$,$N$ and $Q$ on its sides. Also $MPNQ$ is a parallelogram. Drawn by Geogebra So we have

$$\cases{\frac{A - M}{A^* - M^*}= \frac{M - B}{M^* - B^*} \\ \frac{A - P}{A^* - P^*} = \frac{P - D}{P^* - D^*} \\ \frac{D - N}{D^* - N^*} = \frac{N - C}{N^* - C^*} \\ \frac{C - Q}{C^* - Q^*} = \frac{Q - B}{Q^* - B^*} \\ M - Q + N - P = 0 }$$

I don't know how to proceed further.

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  • $\begingroup$ Do you consider a line as a degenerated parallelogram? $\endgroup$ – Anand Aug 27 '20 at 10:40
  • $\begingroup$ @Anand No by a parallelogram I mean a quadrilateral with two pairs of parallel sides. $\endgroup$ – S.H.W Aug 27 '20 at 10:42
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    $\begingroup$ we can inscribe infinite number of parallelograms in a quadrilateral parallel to its diagonals $\endgroup$ – endgame yourgame Aug 27 '20 at 11:31
  • $\begingroup$ @endgameendgame Would you show some examples, please? $\endgroup$ – S.H.W Aug 27 '20 at 11:34
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    $\begingroup$ draw diagonals of of the quadrilateral then draw a line segment parallel to one of the diagonals. connect the line segment with onother line parallel to the diagonals. repeat the process and you get a parallelogram... you can prove this with thales theorem $\endgroup$ – endgame yourgame Aug 27 '20 at 14:23
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Considering you are interested in the following question:

Can we say that a parallelogram is inscribed in a quadrilateral if and only if it is formed by joining the mid-points of sides of the quadrilateral.

The above statement is not true. For instance, consider the following construction:

Draw two squares $ABCD$ and $A'B'C'D'$. Consider the quadrilateral formed by $$\{AA'\cap BB',BB'\cap CC',CC'\cap DD', DD'\cap AA'\}$$This quadrilateral will obviously have at least two distinct parallelogram (more precisely squares).

enter image description here

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  • $\begingroup$ Would you give an image for this construction, please? I couldn't follow that. $\endgroup$ – S.H.W Aug 27 '20 at 11:48
  • $\begingroup$ @S.H.W I've added the image for your kind reference. $\endgroup$ – Anand Aug 27 '20 at 11:52
  • $\begingroup$ Thank you very much. $\endgroup$ – S.H.W Aug 27 '20 at 12:38
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    $\begingroup$ @Anand, you started from squares. Can one start from any given quadrilateral and then find parallelograms or/and squares (apart from the medial parallelogram ofcourse)? $\endgroup$ – cosmo5 Aug 27 '20 at 13:57
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    $\begingroup$ @cosmo we can find many parellelograms in any quadrilateral but in general we cannot inscribe a square in a quadrilateral (eg. rectangle) $\endgroup$ – endgame yourgame Aug 27 '20 at 14:20

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