Prove that if a triangle's area and sides are all integers, its perimeter must be even.

(In reply to complete solution by Daniel)

My solution is similar.

Squaring Heron's formula and substituting s=p/2, we see area squared equals (p/2)(p/2-a)(p/2-b)(p/2-c) or (p/2)(p-2a)/2*(p-2b)/2*(p-2c)/2 or p(p-2a)(p-2b)(p-2c)/16.

Given area is an integer, assuming p is odd leads to a contradition as then p(p-2a)(p-2b)(p-2c) is odd, and area squared wouldn't be an integer. So the perimeter must be even.

Posted by Gamer on 2009-01-16 23:35:09