Minimal triangle via internal point P in an angle


Given an angle in a plane with vertex O and a point P in the interior of the angle. Take a line through P intersecting the sides of the angle at points A and B. Determine a construction for the line to minimize the area of the triangle AOB.


Do you wish to experiment with GSP.      Yes      No

Can you find any special cases? What if P lies on the angle bisector? If P is not on the angle bisector, can you find reasons why the isosceles triangle is not the one with minimal area?

Note: Here is a case where GSP can probably generate pretty good intuition about what line will minimize the area of triangle AOB but not yield much insight on how to construct it or even to prove that it is the line generating the minimal area. Is the line unique?