Let $E:y^2=f(x)=x^3+Ax+B$ an elliptic curve in $\mathbb{Q}$ with the additional hypothesis that all roots $e_1,e_2,e_3$ of $f$ are in $\mathbb{Q}$. I'm trying to prove the fact that:
$P=(a:b:1)\in E(\mathbb{Q})$ is a double point (i.e., there is $Q\in E(\mathbb{Q}$) such that $2Q=P$) if and only if all $a-e_1,a-e_2,a-e_3$ are squares in $\mathbb{Q}$.
I've already proven the direction $\Rightarrow$, but I'm stuck with $\Leftarrow$.
Here is my attempt: if $P$ is a double point, this means there is a point $Q$ such that the line through $-P$ and $Q$ meets $E$ only at these two points, with multiplicity $2$ in $P$. Letting $y=\lambda (x-a)-b$ to be this line, the polynomial $$(\lambda(x-a)-b)^2-f(x)$$
must have one double root $c\neq a$ in $\mathbb{Q}$, and this will give the first coordinate for $Q$.
I know this is supposed to work, but it's mainly brute force, so maybe there is some better way I don't know.