A common approach to proving the Nagell-Lutz theorem is to use a "$p$-adic filtration" $E(\mathbb{Q}_p) \supset E^0 \supset E^1 \supset \dotsb$, which has several properties including that every point in $E^n$ with homogeneous coordinates $(x:y:z)$ must satisfy $y \neq 0$ and $x/y \in p^n\mathbb{Z}_p$.
Next, apply the key lemma (Lemma 5.9 in Milne, and Lemma 9.3.2 in these lecture notes) that for all $P_1,P_2,P_3 \in E(\mathbb{Q}_p)$, with $P_i = (x_i:y_i:z_i)$, $$ P_1 + P_2 + P_3 = O \implies \frac{x_1}{y_1} + \frac{x_2}{y_2} + \frac{x_3}{y_3} \in p^{5n}\mathbb{Z}_p $$
There seems to be a gap in both proof of this lemma that I linked to: they both say that assuming that $P_1 \neq P_2$, the line joining those two points is $z = \alpha x + \beta$, where we have set $y = 1$. Then they compute $v_p(\alpha)$ and $v_p(\beta)$. But shouldn't the line in general be $\alpha x + \beta y + \gamma z = 0$, where $\gamma$ could be zero, failing to give the form $z = \alpha x + \beta$?
Or is there some reason that $\gamma \neq 0$?