Understanding Elliptic Divisibility Sequence (EDS)

Question

I am sorry if this is a basic question but I am pretty new to elliptic curves. More precisely I am trying to understand elliptic divisibility sequence. While I was searching online I came across to:
''Given an elliptic curve $E$ in short Weierstrass form, $E:y^2=x^3+ax+b$ with $a,b \in \mathbb{Z}$, let $P \in E(\mathbb{Q})$ be non-torsion. The shape of the equation forces the expression of the point $P$ to be in the form $P=(\frac{A}{B^2},\frac{C}{B^3})$ where $A,B,C \in \mathbb{Z}$ such that gcd$(AC,B)=1$.
However I don't see why every point of $E$ is of this form. In fact when I tried to place the coordinates of $P$ into $E$ it didn't seem to satisfy the equation and when I found a point of $E$ it wasn't of this form.
Can anyone help me see what I am missing in here?
Thank you in advance