Consider a convex quadrilateral whose diagonals have lengths p,q and whose area is A.
What is the maximum area of a rectangle that circumscribes the given quadrilateral?
For the quadrilateral, A = sin(t)*(pq/2)
where t is the angle between the two diagonals.
If a circumscribing rectangle has a side perpendicular to one of the diagonals, say 'p', then its area is p * (q*sin(t)), which is twice the area of the original quadrilateral.
Also, note that if the circumscribing shape could be a parallelogram instead of a rectangle, and if its sides were all perpendicular to p or q, then that parallelogram's area would also be double the area of the original quadrilateral.
But I haven't proved anything about maximum area.
Posted by Larry
on 2020-01-05 06:49:52