How do we show that equivalence?

Question

We have the triangle ABC. We have that $X\in AC$, $Y\in BC$ and $Z=AY\cap BX$, where $X,Y\neq A,B,C$. I want to show that AB is parallel to XY iff $\frac{|CX|}{|CA|}=\frac{|ZB|}{|ZX|}=1$.

Could you give me a hint what we have to show?

When $AB$ is parallel to $XY$ then we have that then $AZB$ and $ZXY$ are congruent. What do we get from that?

Answer 1

Alternatively: if $CA=CX$ and $ZB=ZX$, then $CZ||AB$. Then $XY \cap AB$ is not empty, so the hypothesis is not valid.

Answer 2

If $\frac{CX}{CA}=1$ then $C$ is a midpoint of $AX$.

Now, since $AB||XY$, we obtain that $ABXY$ is parallelogram and $AY\cap BX=\oslash$