Proposition #1:

All integral triangles have a perimeter that is an even integer

Heron's formula calls for the square root of the product of four values. One of these values is the semi-perimeter and the other values are differences of the semi-perimeter and the sides of the triangle. If the perimeter of the triangle is odd, then the semi-perimeter will not be an integer. Since all of the sides of the triangle are integers, then this means that all of the terms in the product under the square root

will also not be integers. If none of the factors in the product are integers, then the product will not be an integer, and this in turn means that the square root is not an integer. Therefore, any integral triangls will have a perimeter that is an even integer.