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.