#### 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.