KöMaL 2015 B4701

Let $A_1 B_1 C_1 D_1$ be a quadrilateral. For any set of four points $A_n B_n C_n D_n$ already defined for a positive integer $n$ , let $A_{n+1}$ be the centroid of the triangle $B_{n}C_{n}D_{n}$. $B_{n+1}$, $C_{n+1}$ and $D_{n+1}$ are defined analogously, with a cyclic permutation of the points. Show that for any starting quadrilateral, the point sequence $A_n$ only has a finite number of points lying outside the unit circle drawn about the centre of mass of the quadrilateral $A_1 B_1 C_1 D_1$.