Points on elliptic curves

Question

I am learning elliptic curves theorem and I have read in more papers that for two distinct points $P$ and $Q$ there is always point $R$ such that $P+Q+R = \infty$. I know that this point should be unique according to all possible shapes of the curve and intersections (http://en.wikipedia.org/wiki/Elliptic_curve). But is there a simple way how to prove uniqueness of such a point $R$?

Answer 1

A sketch of a proof may look like this:

As what was said in the comments, it is obvious that $P+Q\in E$. So let's just say that $S=P+Q$. Now, since points on an elliptic curve form a group (abelian group), there is an inverse for $S$. Notice that $S$ must satisfy the equation of $E$ to lie on $E$.

So say $E$ is given by the equation $y^2=x^3+ax+b$. Now, since $\mathcal{O}$ is the point of infinity, the inverse of our point $S$ must lie on the line $$x=\text{$x$-coordinate of $S$}.$$

Note that plugging the line into the equation of $E$ will give us at most two solutions, so one solution belongs to $S$, while the other gives us the unique inverse.