What if a line from P3 tangent to w1 is parallel to P1P2?! Call the tangent point p1. Then, p1P3=P1P3/2 and 4^2+P1P3^2/4=P1P3^2.

P1P3^2=16/(1-1/4) and P1P3=8/sqrt(3) and the area is just twice that A=16/sqrt(3), since the height of the triangle is, by our assumption, 4.

Of course, p1P3 may not be parallel to P1P2. So far, I have not found a way to determine that one way or the other. It eyeballs pretty well!

PS: why has the first "correct" answer been deleted??