Points A and B lie in the same plane. How many isosceles right triangles in the plane have A and B as vertices?
E) Infinitely many
That side can be the hypotenuse of such a triangle on either side of it, and it can also be one leg of such a triangle in 4 orientations. Look:
The smaller, darker triangles are where segment AB is the hypotenuse. The larger triangles are the four other orientations.