I've encountered a problem in which I want to maximise the lengths of the sides of a triangular structure while still fitting within an area of 220 x 100. This reduces to maximising the lengths of the sides of triangle with base 220 and height 100.
How could the maximised lengths be calculated?
Edit:- Adding clearer bounds to the problem
The triangle must fit in an area bounded by sides of 300 & 100.
This is not a rigorous proof, but I hope is on the right track. If you have two triangle corners fixed somewhere in the rectangle (incl. the edge), and you are looking for where to put the third one to maximise the perimeter, you are in essence trying to find the largest ellipse with those two corners as focus points, which still intersects the rectangle (and then pick one of the intersecting points). Thus, it is fairly obvious that this ellipse will catch a corner of the rectangle, i.e. the third point must be at the corner of the rectangle. By applying the same argument, you conclude that all three corners of the triangle must coincide with the corners of the rectangle. Thus, your maximum is achieved on a right-angled triangle having two sides of the rectangle as its own sides.