Smallest possible triangle to contain a square I was looking at this stack exchange question* and started thinking about the case of a polygon with 4 sides: a square. 
The question asks for a program that can take a polygon of N sides and return the smallest possible polygon of N-1 sides that contains the original polygon.
Thinking about it, I wasn't sure what the optimal triangle encapsulating a square would look like. In KSab's answers, he returns a triangle that looks like this:

This area is 2*(area of the square). With some easy guess-and-check work I found that this seems to be optimized. How do you prove that there is no better solution?
EDIT: another consideration for the orientation of the triangle could be this:


Apparently this question is fairly old. 
There is a geometric solution mentioned in V. V. Prasolov, Problems in Planimetry II, Nauka, 1986 (16.15). 
I have worked out algebraic solutions for squares, regular pentagons, and regular hexagons (see AMTNYS Journal NYSMTJ Vol. 65, No. 1, 2015, pp.29-33), and also geometric solutions for these polygons using the above reference (to appear).
The results can be generalized to the n-gon case using either method of proof (to appear).
