Let $k$ be a field with $\operatorname{char}(k) \ne 2,3$, let $a,b \in k$, and let
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\begin{align*}
E: y^2 = x^3 + ax + b
\end{align*}
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be a nonsingular affine Weierstrass cubic over $k$. Let $P=(x_1,y_1)$ and $Q=(x_2,y_2)$ be affine $k$-points of $E$.
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Assume either that $P \ne Q$ and the line through $P$ and $Q$ is not vertical, or that $P=Q$ and the tangent line to $E$ at $P$ is not vertical. Let this line be
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\begin{align*}
\ell: y = mx+c
\end{align*}
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for some $m,c \in k$. Let $P*Q=(x_3,y_3)$ denote the third affine intersection point of $\ell$ with $E$, counted with intersection multiplicity, so that in the tangent case $P=Q$ the point $P$ is counted twice. Then