While Euler analysis is correct as far as it goes, the problem is still solvable. Everyone so far has made an assumption about the meaning of the phrasing of the problem which is not required. If a more liberal interpretation is made, the Eulerian difficulty goes away and the problem admits of an easy solution.
(I can't claim to have found the solution myself: this problem has been around for a long time, and so have I!
By the way: Euclid used "line" in exactly the same way that kinnabird5 did. It is only recently that people started demanding that "line" by itself should mean what used to be called a "straight line, extended indefinitely in both directions".