KöMaL 2016 A662

The points $A_1$, $A_2$, $A_3$, $A_4$, $B_1$, $B_2$, $B_3$, $B_4$ lie on a parabola in this order. For every pair $(i,j)$ with $1\le i,j\le4$ and $i\ne j$ , let $r_{ij}$ denote the ratio in which the line $A_jB_j$ divides the segment $A_iB_i$. (That is, if $A_iB_i$ and $A_jB_j$ meet at $X$ then $r_{ij}=\frac{A_iX}{XB_i}$.) Show that if two of the numbers $r_{12} \cdot r_{21} \cdot r_{34} \cdot r_{43}$, $r_{13} \cdot r_{31} \cdot r_{24} \cdot r_{42}$ and $r_{14} \cdot r_{41} \cdot r_{23} \cdot r_{32}$ coincide then the third one is also equal to them.