Four Travelers Problem

geometryfan

(For me, it is pretty hard. Hope you high hands find it interesting.)

Four roads on a plane, each a straight line, are in general position so that no two are parallel and no three pass through the same point. Along each road walks a traveler at a constant speed. Their speeds, however, may not be the same. It's known that traveler #1 met with Travelers #2, #3, and #4. #2, in turn, met #3 and #4 and, of course, #1. Please show that #3 and #4 have also met.

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