Question 8
(GBR 4) Let $S$ be a convex quadrilateral $ABCD$ and $O$ a point inside it. The feet of the perpendiculars from $O$ to $AB,BC,CD,DA$ are $A_1,B_1,C_1,D_1$ respectively. The feet of the perpendiculars from $O$ to the sides of $S_i$, the quadrilateral $A_iB_iC_iD_i$, are $A_{i+1}B_{i+1}C_{i+1}D_{i+1}$, where $i=1,2,3$. Prove that $S_4$ is similar to $S$.