Why is there at most one point that can cut a segment in a given ratio?

Question

I am trying to understand the proof of the Converse Menelaus Theorem.

In the proof, it is written that from $\frac{AF}{FB}=\frac{AF'}{F'B}$, it follows that at most one point can cut a segment in a given ratio. I know it is quite intuitive, but I am still not 100% convinced why there should be at most one point? Why is it not possible for more than one point to cut a segment in a given ratio? I cannot give a counterexample but is there any other more convincing argument?

This maybe a very simple question but many thanks for the helps!

Answer 1

Let $L, R > 0$. Then the only solution to the system $x+y=L, y=Rx$ is $x=L/(R+1), y=RL/(R+1)$ which can be seen by solving the equations.

Answer 2

Hint: The function $ y = \frac{x}{1-x} $ has an inverse function.

Answer 3

You haven't quite stated it correctly. It doesn't say that it follows that one point can cut a line in a given ratio. It says the desired result follows if you use this fact. I think you know this already though.

Let $C$ be on $AB$, and define the ratio $r=AC/CB$. Suppose we allow $C$ to slide towards $A$. Then $r$ will decrease. Conversely if we move $C$ towards $B$ then $r$ will increase. So any change in $C$ is guaranteed to change $r$ with no possibility of it returning to its initial value. So there can only be one point $C$ for any given ratio.