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Let's denote the four straight lines l1, l2, l3, l4 and identify the Travelers by their numbers - #1, #2, #3, and #4. Draw a line perpendicular to the plane in which the four roads are located and think of it as a time axis. Each of the fellows travels with a constant speed. Therefore, the graphs of their motion are straight lines, say, m1, m2, m3, m4.
The fact that point P = (x, y, t) belongs to m(i) is equivalent to saying that
1. point Q = (x, y) lies on l(i).
2. #i passed through the point Q at the time t.
From 1. it follows that projection of mi onto the plane of roads coincides with li.
Also, since #1 and #2 met, at the time of their encounter they were located at the same planar point. Therefore, by 2., m1 and m2 intersect. It must be remembered that two intersecting lines in space define a unique plane. Since #3 met both #1 and #2, m3 intersects both m1 and m2. Therefore, they all lie in the same plane. But the same argument applies to #4 as well. Hence, all four lines m(i), i=1,2,3,4 lie in the same plane. Finally, lines m3 and m4 could not be parallel because their respective projections on the horizontal plane, l3 and l4, intersect. The fact that the lines m3 and m4 intersect means that #3 and #4 happened to be at the same planar point at some point in time which means they have indeed met. |
