Answer: Conjecture: There is no triangle with side lengths N, 2N, and 3N (where N is a positive real number)
Proof:
We prove this by contradiction: Suppose there was an N for which we can construct a triangle with side lengths N, 2N, and 3N. We then apply the triangle inequalities tests. It must hold that:
N + 2N > 3N
3N > 3N
3 > 3
which is False, for any value of N. This means that the original choice of N is not possible. Since the inequality is False for any value of N, there cannot be any triangle with the given side lengths, thus proving our conjecture.
