## 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

Assume the three travelers P, Q, R meet each other. Then either the roads are concurrent (and the three meet at the same point at the same time) or, at all times, P, Q, and R are collinear. |

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

(1) | PQ||P'Q', PR||P'R', QR||Q'R'. |

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

What if applet does not run? |

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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66202007