Let denote the angle

1. The position vector of A is (-r,0)

The position vector of B is (r, 0)

The position vector of P is

2. Then

and

3. Now calculate

And therefore the legs of a right triangle are perpendicular. (Thales theorem)

To the second question:

The cordinates of the squares vertices are O(0, 0), A(s, 0), B(s, s), C(0, s) where s means the side length.

Calculate the vectors of the diagonals and prove that the scalar product of these two vectors is zero.