How do you find the third vertex of a triangle when given two other points and an area?

You cannot. At least, not one point.

If the distance between the two given points is b, and the area of the triangle is A, then let h = 2A/b

Then draw two lines parallel to the line formed by the two given points - one above and one below, each of these lines being at a distance h from the first line.

Any point on either of these two lines will meet your requirements.