Are you a disciple of the Goddess of wisdom?

Logic Level 3

Athena, the Olympiad Goddess of wisdom, had a test for her disciples - Odysseus and Perseus. She told them:

I am thinking of a triangle whose sides are integers.
Perseus, I will tell you its perimeter.
Odysseus, I will tell you its area.
You have to figure out the side lengths of the triangle.

Since they were both vying for Athena’s favor, they were hesitant to reveal too much information, and guarded what they said.

Perseus: The Goddess knows that just the perimeter doesn’t tell me the side lengths.
Odysseus: The Goddess knows that just the area doesn’t tell me what the perimeter is.
Perseus: Athena be praised, the answer has struck me.
Odysseus: Athena be praised, the answer has struck me too.

Now, your task is: how many non-congruent triangles could Athena be thinking of?

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