Question 49
Let $\{A_1, A_2, \ldots, A_{2m+1}\}$ and $\{B_1, B_2, \ldots, B_{2n+1}\}$ be points in the plane such that the $2m + 2n + 2$ lines $A_1A_2, A_2A_3, \ldots, A_{2m}A_{2m+1}, A_{2m+1}A_1, B_1B_2, B_2B_3, \ldots, B_{2n}B_{2n+1}, B_{2n+1}B_1$ are all different and no three of them are concurrent. Show that we can find $i$ and $j$ such that $A_iA_{i+1}$ and $B_jB_{j+1}$ are opposite sides of a convex quadrilateral (if $i = 2m+1$, then we take $A_{i+1}$ to be $A_1$. Similarly for $j = 2n+1$).