Four Travelers Problem

Let four Lines in a Plane represent four roads in General Position, and let one traveler $T_i$ be walking along each road at a constant (but not necessarily equal to any other traveler's) speed. Say that two travelers $T_i$ and $T_j$ have ``met'' if they were simultaneously at the intersection of their two roads. Then if $T_1$ has met all other three travelers ($T_2$, $T_3$, and $T_4$) and $T_2$, in addition to meeting $T_1$, has met $T_3$ and $T_4$, then $T_3$ and $T_4$ have also met!

References

Bogomolny, A. ``Four Travellers Problem.'' http://www.cut-the-knot.com/gproblems.html.