Spies Passing in the Night - Solution

The crucial thing needed to solve this problem is the idea that we can map the positions of the two armies, the two spies, and the traveller, against time, and get five straight lines. The slopes of these lines will of course be the various speeds. This is exactly the setup covered by Pappus' Theorem.

Now the result just falls out - it's easier to describe with a diagram than with words. (Thanks to Ashish Bhambhani from Nagpur in India for his correct solution and the diagram below.)