Consider a person P who won the greatest number of matches. (If there is a tie for most, choose any of them.) This person is a dominant player.
If P is not a dominant player, then some person Q beat P and every person P beat. But this is more people than P beat, contradicting the choice of P as the player with the most victories.
As many have mentioned, there is no "inside the box" answer to the problem.
Depending on the exact statement of the problem, there may be trick answers. For example, as stated, the lines may not touch, but there is no rule against a line going through another circle in the middle of its route.