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Considering the "visual" method of explaining elliptic curve addition, I wonder what you are supposed to do if one of the points being added is tangent to the curve?

With the classic shape e.g. $y^2 = x^3 - x + 2$ , you could get a horizontal line, that only touches the curve twice, not three times:

enter image description here

The third point can't be $0$, because that would mean that $B=-A$, clearly not the case. It can't be $B$ either, because that would mean that $A=0$ ?

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  • $\begingroup$ Welcome to MSE. Please use MathJax. $\endgroup$ Commented Oct 16, 2017 at 8:48
  • $\begingroup$ How can a point be tangent to a curve? $\endgroup$
    – Bernard
    Commented Oct 16, 2017 at 8:50

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In this case, we would say that your straight line passes through the points: $$ [{\rm Blue}], \ \ [{\rm Red } ], \ \ [{\rm Red}].$$ Notice that I've deliberately included the point $[{\rm Red } ]$ twice in this list! This is because the straight line intersects the elliptic curve with multiplicity two at the point $[{\rm Red } ]$.

The rest is standard: having identified these three(!) intersections points between the straight line and the elliptic curve, we deduce the group addition rule: $$ [{\rm Blue}] + [{\rm Red}] = - [{\rm Red}]$$

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    $\begingroup$ Very clear, thanks Kenny! $\endgroup$ Commented Oct 16, 2017 at 9:26

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