If I start with an equilateral triangle 10 miles on each side (area ~43 square miles), then move one of the corners 10 miles in order to maximize the new area, it’s obvious what I should do: move one of the corners 10 miles away from the centre of the original triangle. This would make my new territory an isosceles triangle with angles 75, 75, 30 (area ~93 square miles).
If instead I can move all three of the corners by 10 miles, I would move each of them 10 miles directly away from the triangle centre and get a larger equilateral triangle (area ~323 square miles).
If however, I could only move two corners, each by 10 miles, moving them directly away from the original centre of the circle, (giving an area of ~187 square miles and angles of 45,45,90), would not in fact give the maximum area.
It is possible to achieve a maximum area of over 191 square miles (moving only two corners 10 miles each), but what would the angles of the resulting triangle be in this case?
