Isoperimetric Theorem for Parallelograms
What Might This Be About?
2 March 2014, Created with GeoGebra
Rectangle among Parallelograms
For any parallelogram (which is not a rectangle) there is a rectangle of the same perimeter but larger area.
Indeed, according to Euclid I.35, parallelograms with the same base and in the same parallels have equal areas. Restricting to rectangles, it is clear that, among rectangles with the same base, the area grows with the distance between the parallel lines. Since in a circle, the point farthest from a diameter, lies on the perpendicular radius, the area of a rectangle among all parallelograms with the same sides is the largest.
The above has been communicated to me in private correspondence by Sidney Kung on 1 March, 2014.