RU-ALLRUSSIAN 1971 Problem 2

(1) $\mathrm{A_1A_2A_3}$ is a triangle. Points $\mathrm{B_1}$ , $\mathrm{B_2}$ , $\mathrm{B_3}$ are chosen on $\mathrm{A_1A_2}$ , $\mathrm{A_2A_3}$ , $\mathrm{A_3A_1}$ respectively and points $\mathrm{D_1}$ , $\mathrm{D_2}$ $\mathrm{D_3}$ on $\mathrm{A_3A_1}$ , $\mathrm{A_1A_2}$ , $\mathrm{A_2A_3}$ respectively, so that if parallelograms $\mathrm{A_iB_iC_iD_i}$ are formed, then the lines $\mathrm{A_iC_i}$ concur. Show that $\mathrm{A_1B_1}$ : $\mathrm{A_2B_2}$ : $\mathrm{A_3B_3} = \mathrm{A_1D_1}$ : $\mathrm{A_2D_2}$ : $\mathrm{A_3D_3}$ .

(2) $\mathrm{A_1A_2\ldots A_n}$ is a convex polygon. Points $\mathrm{B_i}$ are chosen on $\mathrm{A_iA_{i + 1}}$ (where we take $\mathrm{A_{n + 1}}$ to mean $\mathrm{A_1}$ ), and points $\mathrm{D_i}$ on $\mathrm{A_{i - 1}A_i}$ (where we take $\mathrm{A_0}$ to mean $\mathrm{A_n}$ ) such that if parallelograms $\mathrm{A_iB_iC_iD_i}$ are formed, then the n lines $\mathrm{A_iC_i}$ concur. Show that $\prod \mathrm{A_iB_i} = \prod \mathrm{A_iD_i}$ .