# 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.

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

R exists in F Proof

.pdf

Download PDF • 66KB

## Visual Examples

Examples of P, Q and R defined in the Real numbers - for reference while, reading the a proof above.

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