There are finitely many lines in a plane $\mathbb{R}^2$. $\textbf{None of them is parallel}$. For each line $L$ which intersects with other lines say at $A_1$, $A_2$, $\cdots$ , $A_n$ in the consecutive order, it is called (1) well-divided (w.d.) if $A_1A_2 = A_2A_3 = \cdots = A_{n-1}A_n$; (2) geometrically-divided (g.d.) if $A_{i+1}A_{i+2}/A_iA_{i+1}=k$ for all $i$. (3) monotone if $A_iA_{i+1}$'s are monotone increasing or decreasing.
For (2) and (3) they must be in the consecutive order.
A well-divided line is trivial if there are only one or two intersections. Similarly, g.d. lines and monotone lines are trivial if there are three intersections or less.
Recently I proved the following result:
Theorem. If all lines are well-divided, then there is at most one non-trivial well-divided line.
My questions are, what if all lines are g.d. lines with the same ratio $k$? What if all lines are g.d. lines with arbitrary ratios? What if all lines are monotone?
I only have some partial results.
Has anyone considered these problems before? Can anyone give me some references about these problems? Thanks.
