Proposition #3:

For all triangles having a semi-perimeter that equals a perfect square >4, there is at least one integral triangle

Consider again the formula for the area of the triangle

if the semi-perimeter is a perfect square, then the square root of s is an integer and the remaining focus is on the square root

Because s > 4, there is some value a for which (s - a) = 4, so the square root of (s - a) is an integer, 2, and the remaining focus is on the square root

When b = c, this becomes the square root of (s - b) sqaured, which is (s - b). Therefore there is at least one integral triangle which is isoceles.