perplexus.info :: Shapes : Triangle Perimeter and Semi-Perimeter

Triangle Perimeter and Semi-Perimeter (Posted on 2015-01-04)

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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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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