Let P be a point in the plane and suppose that it is white. (If it is blue the argument is still valid, just interchange white and blue.) Construct a circle of radius pi and center P. Let Q be some point on the circumference of the circle. If Q is white then P and Q are the points required. If Q is blue construct a circle of radius pi and center Q. Let R be one of the two points where the circles intersect.
If R is white then P and R are the desired points. If R is blue then Q and R arre the desired points.
An interesting feature of this problem is that it is true for any number, not just pi. What I mean is that if A is any positive number then there are two dots of the same color paint are exactly A feet apart.Cheers,