# 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 definednot 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.

Click the download button on the right to download and view:

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