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Triangle Perimeter and Semi-Perimeter (Posted on 2015-01-04) Difficulty: 3 of 5
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.

No Solution Yet Submitted by K Sengupta    
Rating: 4.0000 (1 votes)

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Solution Part 2 proof (spoiler) | Comment 2 of 3 |
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.

  Posted by Steve Herman on 2015-01-04 11:57:09
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