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 lP||PR through P' and similarly lQ||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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