Can someone please help me prove this geometry question?

[eddy123]

I have been tackling this geometry question for some time now, but I can't seem to figure anything out. The question is:
"Prove that each convex quadrilateral can be cut into four parts, so that a parallelogram can be drawn from these parts.
I have tried cutting through diagonals, midpoints etc., but I can't seem to get a parallelogram.

If anyone has any ideas, I would really appreciate your help.
Thank you!

 

[lex]

I'm not exactly clear what you are being asked to do, but it may be this (which claims to be a construction which cuts four parts off a convex quadrilateral to leave an inscribed rhombus):

Inscribing a rhombus within a convex quadrilateral

I was wondering if it is possible to inscribe a rhombus within any arbitrary convex quadrilateral using only compass and ruler? If possible, could you describe the method? If not could you give an

math.stackexchange.com

 

[lex]

Using the midpoints, you can draw a parallelogram, more easily:

Varignon's theorem - Wikipedia

en.wikipedia.org

(if this is the problem you're being asked to solve)!

 

[Dr.Peterson]

I have been tackling this geometry question for some time now, but I can't seem to figure anything out. The question is:
"Prove that each convex quadrilateral can be cut into four parts, so that a parallelogram can be drawn from these parts.
I have tried cutting through diagonals, midpoints etc., but I can't seem to get a parallelogram.

If anyone has any ideas, I would really appreciate your help.
Thank you!

It isn't clear what "a parallelogram can be drawn from these parts" means. I suspect you may mean a dissection -- that the four parts can be rearranged to form a parallelogram.

It may help if you tell us the context of the question -- what theorems do you have, and what topics have you been studying that may be of use?

My own first thought is related to tiling the plane with the quadrilateral, and then overlaying a tiling of parallelograms on it. Even if you have not been studying tilings, that may suggest how to do the dissection, which can then be proved valid.

 

[eddy123]

It isn't clear what "a parallelogram can be drawn from these parts" means. I suspect you may mean a dissection -- that the four parts can be rearranged to form a parallelogram.

It may help if you tell us the context of the question -- what theorems do you have, and what topics have you been studying that may be of use?

My own first thought is related to tiling the plane with the quadrilateral, and then overlaying a tiling of parallelograms on it. Even if you have not been studying tilings, that may suggest how to do the dissection, which can then be proved valid.

Yes, like imagine the quadrateral is a piece of paper. I have to cut the paper into 4 parts, then rearrange those parts to form a parallelogram. You can turn and twist those parts, but they can't overlap or leave gaps in the middle.

This is not something I am learning, this is just one Olympiad question and I found it interesting, but couldn't solve it. Although it would be good to have an answer before next week. I think for this I have to use the parallelogram features, like if the opposite edges and equal and parallel, then its a parallelogram and so on.

I have gotten close - like If you cut through one diagonal and 2 opposite midpoints, or through 2 side edge midpoints, the 2 other side edge midpoints, then thought opposite midpoints, I got really close to a parallelogram, but not exactly.

 

[Dr.Peterson]

It sounds like you have some good ideas, so you may not need mine. Here is the tiling I mentioned, which can be done with any quadrilateral, and results in a parallelogrammish figure, which can itself be repeated over and over to tile the whole plane:



That may help you, or may get in the way -- it took me a while to figure out what would work, even though I had the basic idea and have seen something like it before.