Construction of a parabola

A parabola is the set of points equidistant from a given point, called the focus, and a given line called the directrix. In the diagram below, F is the focus and AB is the directrix. C is any point on the directrix.
You can play around with A, B, F and C. Given C and F, how do we find the point P on the parabola equidistant from C and F? Hint: consider the right-angled triangle FGP; from this you may find the radius, |FP|. If Q is the other point of intersection of the two circles, why does it lie on the axis, DF, of the parabola? What do you notice about the line PQ? Can you convince first a friend and then a sceptic about your answers to these questions?