= Solution
The <projective completion> has identity $O=[0:1:0]$. For this <Weierstrass equation of an elliptic curve>, negation is
$$
-(x,y)=(x,-y-1).
$$
For distinct affine points $P,Q$, draw their chord; for $P=Q$, use the tangent with its <intersection multiplicity>. If the third intersection is $R$, define $P\oplus Q=-R$. A vertical chord pairs $P$ with $-P$ and gives their sum $O$; also $P\oplus O=P$. A vertical tangent at a point satisfying $2y+1=0$ gives $2P=O$.
For a nonvertical line $y=mx+b$, substitution into the cubic gives
$$
x^3+(4-m^2)x^2+(-2-2mb-m)x-(b^2+b)=0.
$$
Thus if $P=(x_1,y_1)$ and $Q=(x_2,y_2)$, the sum has coordinates
$$
x_3=m^2-4-x_1-x_2,\qquad y_3=-mx_3-b-1.
$$
Here $m=(y_2-y_1)/(x_2-x_1)$ for a chord, and <implicit differentiation> gives
$$
m=\frac{3x_1^2+8x_1-2}{2y_1+1}
$$
for a nonvertical tangent. These formulas include the repeated-root case. <Associativity> can be seen without a lengthy coordinate calculation: a line cuts the cubic in a divisor equivalent to $3(O)$, and $P\mapsto[(P)-(O)]$ identifies the curve with its degree-zero <divisor class group>. The chord-and-tangent operation becomes addition of <divisor classes>, hence is associative.
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