Three Roads, Three Travelers

The applet below provides an alternative demonstration to a lemma crucial in a solution to the Four Travelers problem. The lemma reads

Let's agree that P, Q, R denote the positions of the travelers at some time t, while P', Q', R' correspond to another time t = t'. The three roads, being in general position, intersect at three points A, B, C. These are the rendezvous points of the travelers. In the applet, triangle ABC and its vertices are draggable, as are the green lines that join the travelers. The green lines, if dragged close to their midpoint, remain parallel to the starting position; for, as we know, the lines joining the travelers remain parallel:

The applet suggests why (1) is impossible, unless the points P, Q, R (and, of course, P', Q' and R' also) are collinear.

In geometric terms, let triangle PQR be inscribed in triangle ABC, with P, Q, R on BC, AC, and AB (or their extensions), respectively. Let P' and Q' lie on BC and AC so that PQ||P'Q'. Consider line l_P||PR through P' and similarly l_Q||QR through Q'. For P', Q' different from PQ, the two lines can't meet on AB because, relative to PR and QR, they moved in different directions.

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