I'm reading through Milne's notes on Elliptic Curves, and going through the proof of the Nagell–Lutz theorem. For that, Milne first tries the prove the following proposition:
Let $P=(x_1:y_1:1) \in E(\mathbb{Q})$, If $P$ and $2P$ have integer coordinates (setting $z=1$), then either $y_1=0$ or $y_1\ \vert\ \Delta$
The proof starts with him assuming $y_1 \neq 0$ and letting $2P= (x_2,y_2:1$). Then he claims $2P$ is the second point of intersection of the tangent line at $P$ with the affine curve $Y^2 = X^3 + aX + b$
The bolded statement is what I don't understand, isn't the second point of intersection $-2P$? Since by the group law we'll have, if $Q$ is the point of intersection, $P + P + Q = O \implies Q = -2P$
Can anybody help me with this?