Question 7
(1) $A_1A_2A_3$ is a triangle. Points $B_1$, $B_2$, $B_3$ are chosen on $A_1A_2$, $A_2A_3$, $A_3A_1$ respectively and points $D_1$, $D_2$, $D_3$ on $A_3A_1$, $A_1A_2$, $A_2A_3$ respectively, so that if parallelograms $A_iB_iC_iD_i$ are formed, then the lines $A_iC_i$ concur. Show that $A_1B_1 : A_2B_2 : A_3B_3 = A_1D_1 : A_2D_2 : A_3D_3$.
(2) $A_1A_2\ldots A_n$ is a convex polygon. Points $B_i$ are chosen on $A_iA_{i + 1}$ (where we take $A_{n + 1}$ to mean $A_1$), and points $D_i$ on $A_{i - 1}A_i$ (where we take $A_0$ to mean $A_n$) such that if parallelograms $A_iB_iC_iD_i$ are formed, then the $n$ lines $A_iC_i$ concur. Show that $\prod A_iB_i = \prod A_iD_i$.
(2) $A_1A_2\ldots A_n$ is a convex polygon. Points $B_i$ are chosen on $A_iA_{i + 1}$ (where we take $A_{n + 1}$ to mean $A_1$), and points $D_i$ on $A_{i - 1}A_i$ (where we take $A_0$ to mean $A_n$) such that if parallelograms $A_iB_iC_iD_i$ are formed, then the $n$ lines $A_iC_i$ concur. Show that $\prod A_iB_i = \prod A_iD_i$.