RU-ALLRUSSIAN 1987 Problem 8

$\mathrm{A_1,A_2,\ldots,A_{2m + 1}}$ and $\mathbf{B}_1,\mathbf{B}_2,\ldots ,\mathbf{B}_{2n + 1}$ are points in the plane such that the $2\mathrm{m} + 2\mathrm{n} + 2$ lines $\mathrm{A_1A_2,A_2A_3,\ldots,A_{2m}A_{2m + 1},A_{2m + 1}A_1,B_1B_2,B_2B_3,\ldots,B_{2n}B_{2n + 1},B_{2n + 1}B_1}$ are all different and no three of them are concurrent. Show that we can find i and j such that $\mathrm{A_iA_{i + 1}}$ $\mathrm{B_iB_{j + 1}}$ are opposite sides of a convex quadrilateral (if $\mathrm{i} = 2\mathrm{m} + 1$ , then we take $\mathrm{A_{i + 1}}$ to be $\mathrm{A_1}$ . Similarly for $\mathrm{j} = 2\mathrm{n} + 1$ ).