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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 1 proof (spoiler) | Comment 1 of 3
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.

  Posted by Steve Herman on 2015-01-04 11:37:26
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