# Inscribed parallelogram in a quadrilateral

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. 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.

• Do you consider a line as a degenerated parallelogram? – Anand Aug 27 '20 at 10:40
• @Anand No by a parallelogram I mean a quadrilateral with two pairs of parallel sides. – S.H.W Aug 27 '20 at 10:42
• we can inscribe infinite number of parallelograms in a quadrilateral parallel to its diagonals – endgame yourgame Aug 27 '20 at 11:31
• @endgameendgame Would you show some examples, please? – S.H.W Aug 27 '20 at 11:34
• 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 – endgame yourgame Aug 27 '20 at 14:23

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).

• Would you give an image for this construction, please? I couldn't follow that. – S.H.W Aug 27 '20 at 11:48
• @S.H.W I've added the image for your kind reference. – Anand Aug 27 '20 at 11:52
• Thank you very much. – S.H.W Aug 27 '20 at 12:38
• @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)? – cosmo5 Aug 27 '20 at 13:57
• @cosmo we can find many parellelograms in any quadrilateral but in general we cannot inscribe a square in a quadrilateral (eg. rectangle) – endgame yourgame Aug 27 '20 at 14:20