Consider a triangle ABC each of whose sides and area is an integer.
Prove separately each of these assertions:
(i) Perimeter of triangle ABC is always even.
(ii) Semi-perimeter of triangle ABC is always composite.
From Heron's formula again,
Area = sqrt(s*(s-a)*(s-b)*(s-c))
We determined in part 1 that s is an integer (because the perimeter is even).
Assume s is prime.
Then, s-a and s-b and s-c are all integers, necessarily less than s, and therefore relatively prime to s.
Therefore, the square root cannot be rational, because the prime factorization of s*(s-a)*(s-b)*(s-c) contains s raised to the first power.
So our initial assumption is wrong, and s must be composite.