Why is the length of one of the segments negative in the Menelaus' theorem? Aren't all distances by definition positive?
I think we all know the Menelaus' theorem, which claims that the following equality in the triangle given below holds:
And as you can see it's implied that one of the segments is negative. Normally, talking about lengths in general, $PQ=QP$, but this case shows that it's actually $PQ=-QP$, while the definition of distance says that $PQ=QP$. There's something I don't understand here - doesn't it contradict the definitions of geometry? Some explanation would be appreciated. Thanks.
When you go from $A$ to $F$ you have to come back from $F$ to $B$ so the segments $AF$ and $FB$ have opposite orientations.
This theorem uses signed distances rather than absolute ones, and this is quite common in this kind of exercise. It distinguishes between the point $F$ external to $AB$ and the internal point for which the absolute ratio would be the same (but positive). For this theorem you need the external point, and you need the signed distance to pick it out unambiguously.