# Elliptic Curve addition: Tangent Point

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: 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 $A=0$ ?

• Welcome to MSE. Please use MathJax. – José Carlos Santos Oct 16 '17 at 8:48
• How can a point be tangent to a curve? – Bernard Oct 16 '17 at 8:50

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: $$[{\rm Blue}] + [{\rm Red}] = - [{\rm Red}]$$