Let an elliptic curve $ E(a,b )$ $$E(a,b ) = \{(x,y)\,|\,y^2=x^3+ax + b\}$$
Where the points of the line $xm + n$:
$$P =(x_1, y_1),\, Q =(x_2, y_2),\, R =(x_3, y_3) \in E(a, b)$$
How can you calculate the product between them? $$Q\cdot P =\, ?$$
Say we have the line $(x_0+ct,y_0+dt)$ which intersects the curve in 3 points : at $t=-1,t=1$ and $t= ?$
Finding $?$ means solving the 3rd root of the cubic polynomial $$P(t) = (y_0+td)^2- (x_0+tc)^3-a(x_0+tc)-b$$ And obtain the factorization $$P(t) = -c^3 (t+1) (t-1)(t-?)$$ Therefore $$? = \frac{P(0)}{c^3}=\frac{y_0^2-x_0^3-ax_0-b}{c^3}$$ From which we obtain the group law $$(x_0-c,y_0-d)+(x_0+c,y_0+d) = (x_0+\frac{y_0^2-x_0^3-ax_0-b}{c^2}, -y_0-d\frac{y_0^2-x_0^3-ax_0-b}{c^3})$$