#### Proposition #4:

#### For all triangles with a semi-perimeter > 9 having perfect
square factors >4, there is more than one integral triangle
having that perimeter

Consider again the formula for the area of the triangle

the specifications for the semi-perimeter indicate that for
some value a which equals s - x, where x is the result when s
is divided by one of its perfect square factors, then the product
s*(s - a) will be equal to s*x which equals a perfect square times
x squared. This leaves the focus on the square root

Since s > 9, then b and c could be determined such that
(s - b) = 9 and (s - c) = 4 and therefore their product is a perfect
square. This cannot be proven completely without considering that
the value for a already determined and the necessary values for
b and c will satisfy the triangle inequality.

The larger size of s also allows for more opportunties when
b = c and therefore there are integral isoceles triangles.