Theorem / Proof: A line connecting P and Q always intersects a third point
How do we know that a line connecting P and Q will always intersect another point on the curve - thus allowing us to define a Group Law for an Elliptic Curve over a given Field?
Please download the following PDF, the proof of:
NOTE: there are more succinctly expressed theorems that take the "point at infinity" for an elliptic curve into account, and can therefore also talk about vertical lines (where x_p = y_q) when P and Q are distinct). However, at this point in the process of the examination of elliptic curves it seems more informative to consider the case defined purely in the affine plane (without considering the point at infinity, or the projective plane) to see how far this definition can be taken, within that construct.
The outcome of avoiding the concept of the "point at infinity", is: we cannot strictly define a Group in the affine plane, because
some points on the curve, when added to each other, do not exist in the field on which the curve defined
not all points on the curve have an additive inverse
Understanding the difference between an elliptic curve defined strictly on the affine plane, and an elliptic curve defined in a way that includes "the point at infinity" is essential understanding why this definition is used, and why this "point at infinity" is seen as the "additive neutral element" of the field.
But for now, these ideas will be simplified, and we will ignore the case in which the slope of the line between two points on the curve is vertical.
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