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Let $E/\mathbb{Q}$ be an elliptic curve. When we add two points on an elliptic curve, we take the line joining them, take the third intersection point and then reflect the point and use that as the sum (at least in the generic case).

Why do we reflect the point? Is there a good reason? Is there a good low-level explanation or does it require some higher level algebraic geometry explanation? I don't think the group law itself suffers if we don't flip the point correct?

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  • $\begingroup$ In the general case, if you want the point at infinity to be zero, then $0+P$ would be $-P$ if you didn't reflect the point. $\endgroup$ – rogerl Oct 26 '15 at 1:26
  • $\begingroup$ Okay great that works. Thanks! $\endgroup$ – user283918 Oct 26 '15 at 14:19
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This rule can be applied when you assume that the zero element is the point of your elliptic curve at infinity: More precisely, when the equation of the affine part has the form $y^2=(x-\alpha_1)(x-\alpha_2)(x-\alpha_3)$. It says that $P+Q+R=0$ if they lie on a line in the generic case. This gives you a geometric way to find $-(P+Q)$. For a generic finite point $P$ the line passing through $P$ and the point at infinity should intersect the curve at $-P$, which is the reflection point with respect to the $x$-axis.

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