Let y =x2/(4p) be the equation of the blue parabola in the figure (the red circle is the circle of curvature at the vertex). Then we have

y' = x/(2p),   y'' = 1/(2p),   y'(0) = 0,   and   y''(0) = 1/(2p).

Thus, at the vertex (0,0) the curvature K is given by

K = |y''(0)| / (1+[y'(0)]2)3/2 = 1/(2p).

It follows that the radius of curvature r at the vertex is

r = 1/K = 2p,

which is what we wanted to prove.

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