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.
Let a,b,c be the three sides.
Perimeter = p = a+b+c
Assume p is odd.
Then s = semi-perimeter = p/2 is not an integer
From Heron's formula,
Area = sqrt(s*(s-a)*(s-b)*(s-c))
Substitute s = p/2, giving
Area = sqrt(p*(p-2a)*(p-2b)*(p-2c))/4
Each of the terms in the square root is odd, so this cannot evaluate to an integral area.
So our initial assumption is wrong, and the perimeter is even.