= Symmetric-coordinate identity for elliptic-curve addition
{title2=$(s_2-a)^2=4s_1(s_3+b)$}
On $y^2=x^3+ax+b$ in <characteristic of a field> other than two, take the <elementary symmetric polynomials> $s_i$ of $x(P),x(Q),x(P+Q)$. If the line through $P,Q$ is $y=\lambda x+\mu$, its third intersection is $-(P+Q)$, so coefficient comparison gives $s_1=\lambda^2$, $s_2=a-2\lambda\mu$, $s_3=\mu^2-b$. Hence $(s_2-a)^2=4s_1(s_3+b)$. This is an identity of <rational functions>; tangent cases follow by specialization, and coordinates at infinity are handled through the projective addition map.
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