The ultimate metapuzzle

If you have been following this blog for a while, you already know of my fascination with metapuzzles. I have recently found one of the most beautiful examples of these riddles in this August’s edition of Ponder This, and following the advice of Travis, I will leave it completely open for the readers to give it a go. Enjoy!

A mathematician wanted to teach his children the value of cooperation, so he told them the following:

«I chose a secret triangle for which the lengths of its sides are all integers.

To you my dear son Charlie, I am giving the triangle's perimeter. And to you, my beloved daughter Ariella, I am giving its area.

Since you are both such talented mathematicians, I'm sure that together you can find the lengths of the triangle's sides.»

Instead of working together, Charlie and Ariella had the following conversation after their father gave each of them the information he promised.

Charlie: "Alas, I cannot deduce the lengths of the sides from my knowledge of the perimeter."

Ariella: "I do not know the perimeter, but I cannot deduce the lengths of the sides from just knowing the area. Maybe our father is right and we should cooperate after all."

Charlie: "Oh no, no need. Now I know the lengths of the sides."

Ariella: "Well, now I know them as well."

Find the lengths of the triangle’s sides.