associativity on elliptic curves -- Milne's proof In the proof that the group law on an Elliptic curve is associative, Milne (http://www.jmilne.org/math/Books/ectext5.pdf, page 28) sets up 3 cubics,
and claims that they all contain the $8$ points $O,P,Q,R,PQ,QR,P+Q,Q+R$ where $AB$ denotes the third point of intersection on $L(A,B) \cap C$. His cubics are


*

*$C = 0$ ($C$ is our elliptic curve.

*$L(P,Q) \cdot L(R,P+Q) \cdot L(QR,O) = 0$
and
3. $L(P,QR) \cdot L(Q,R) \cdot L(P,O) = 0$
where $L(A,B)$ is the (projective) line determined by $A$ and $B$.
I get why the first two points contains all $8$ points. Why does the last one contain all of them? Maybe there's a typo somewhere, and he meant some other cubic for 3?
edit 1: so obviously $O,P,Q,R,QR$ are on the third line. Why are the other $3$ points on that cubic?
edit 2: I'm starting to think that there is a typo in Milne's book. Either way, I have found another line containing all $8$ points, so I guess it doesn't matter.

I agree that this looks like a typo – or even two. Consider the illustration on that same page:

Apparently the last cubic should be
$$L(P,Q\color{red}{+}R)\cdot L(Q,R)\cdot L(P\color{red}{Q},O)=0$$
or something along these lines. It corresponds to the three horizontal lines in that illustration, just like the second cubic correctly corresponds to the vertical ones.
The errata document does not mention this problem. You might consider contacting the author about this.
