Confusion about the Thales theorem

Question

We have a $△ABC$. Points $M,N$ lie on $AB$ and $AC$. $AM=10$ ; $AB=30$ ;$AN=6$; $AC=18$. The Thales theorem says that if $\frac{AM}{AB}$ = $\frac{AN}{AC}$, then $BC||NM$. If I sovle it, we indeed find that $\frac{10}{30}$ = $\frac{6}{18}$ = $\frac{1}{3}$ so $\frac{AM}{AB}$ = $\frac{AN}{AC}$. My confusion is that for the last year, I have been solving it as $\frac{AM}{MB}$ = $\frac{AN}{NC}$ and have always been able to get the correct answer for weather BC||NM. I have tried to prove it using an equation but am not able to for the life of me figure out why this works.

Answer 1

$$\begin{align}\frac {AM}{MB} &= \frac {AN}{NC} \\\iff\quad\quad\quad \,\,\,\frac {MB}{AM} &= \frac {NC}{AN} \\\iff\quad\frac {AM+MB}{AM} &= \frac {AN+NC}{AN} \\\iff\quad\quad\quad \,\,\,\frac {AB}{AM} &= \frac {AC}{AN} \\\iff\quad\quad\quad \,\,\,\frac {AM}{AB} &= \frac {AN}{AC}\end{align}$$

although both versions can be proved via similar triangles.