Area Equals Perimeter: Heronian

Number Theory Level 4

$ABC$ is a right triangle with integer side lengths. The area of $\triangle ABC$ is equal to its perimeter.

What is the sum of all possible areas of $\triangle ABC$ ?

This problem is a part of the set "I Don't Have a Good Name For This Yet". See the rest of the problems here. And when I say I don't have a good name for this yet, I mean it. If you like problems like these and have a cool name for this set, feel free to comment here.