Putnam 1946 B2

Let $A, B$ be two variable points on a parabola $P_{0}$ , such that the tangents at $A$ and $B$ are perpendicular to each other. Show that the locus of the centroid of the triangle formed by $A,B$ and the vertex of $P_0$ is a parabola $P_1 .$ Apply the same process to $P_1$ and repeat the process, obtaining the sequence of parabolas $P_1, P_2 , \ldots, P_n$ . If the equation of $P_0$ is $y=m x^2$ , find the equation of $P_n .$