# The smallest parallelogram that contains a convex quadrilateral

I try to find the smallest parallelogram in terms of area that contains a convex quadrilateral(A,B,C,D).

I am pretty sure it must be constructed from two neighboring sides of the quadrilateral.

But which ones?

I have no approach for a condition like "smallest angle" or "shortest distance" ect.

• What is meant by "smallest parallelogram"? The one with least area? The one with least diameter (as a set)? The one with least perimeter? Commented Feb 22, 2017 at 3:34
• area, thanks for pointing that out Commented Feb 22, 2017 at 3:38
• It seems someone has already worked this out here.
– Jens
Commented Feb 22, 2017 at 5:41
• @Jens thank you! But if I understand the paper correctly it "just" tries different parallelograms and is efficient for polygons with 5 or more vertexes. I hope there is a simpler solution for my case with only convex quadrilateral. Commented Feb 22, 2017 at 15:37

If no sides are parallel, find a side $AB$ whose two vertices have interior angles whose sum is less than $180$ degrees. One of the two adjacent sides, say $BC,$ must lie on an MEP, and a brief inspection confirms that there is an MEP with one side equal to $AB$ and the other containing $BC.$
Therefore a side whose interior angles sum to less than $180$ degrees may be used as one edge of an MEP. One of the adjacent sides will also have interior angles whose is less than $180$ degrees; it is the other side of an MEP.
Short version: find three vertices of the convex quadrilateral such that the sum of interior angles of either adjacent pair of vertices is no greater than $180$ degrees. Those three vertices are vertices of an MEP.