Group law on an elliptic curve

Remarks

Geometrically, $P + Q + R = O$ exactly when $P$, $Q$, $R$ are the three intersection points, counted with multiplicity, of $E$ with a line in the projective plane. The number $\lambda$ is the slope of the line through $P$ and $Q$, or of the tangent at $P$ when $P = Q$.

Example: on $y^2 = x^3 + 1$ over $\Q$, take $P = Q = (2, 3)$. Then $\lambda = \frac{3 \cdot 4 + 0}{6} = 2$, $x_3 = 4 - 2 - 2 = 0$ and $y_3 = 2(2 - 0) - 3 = 1$, so $2P = (0, 1)$, and indeed $1^2 = 0^3 + 1$.

Non-example: the operation $P * Q$ that sends $P, Q$ to the third intersection point of their line with $E$, without the reflection, is not a group law. A neutral element $e$ for $*$ would have to lie on the tangent line at every point of $E$. But only finitely many tangent lines of a smooth cubic pass through a given point.

Associativity is the hard part of the proof that $E(K)$ is a group; it follows from the theorem of Riemann–Roch or from a direct computation with the formulas. For a general Weierstrass equation the formulas change, but the geometric rule stays the same.