To Cooperate or Not?

I got this puzzle from IBM website. They have many more mind boggling puzzles out there. Actually they post it on monthly basis and the one given below is the latest puzzle on their site for August 2011.

Tricky Triangle Riddle

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 edges.”

Ariella: “Well, now I know them as well.”

Can you find the lengths of the triangle’s sides and explain the dialog above?

Source: IBM

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