I'm trying to calculate the expected area of a random triangle with a fixed perimeter of 1.
My initial plan was to create an ellipse where one point on the ellipse is moved around and the triangle that is formed with the foci as the two other vertices (which would have a fixed perimeter) would have all the varying areas. But then I realized that I wouldn't account for ALL triangles using that method. For example, an equilateral triangle with side lengths one third would not be included.
Can anyone suggest how to solve this problem? Thanks.