= Solution
{c}
Work with the smooth projective cubic, including its points at infinity. For two points $P,Q$, draw their chord, using the tangent when they coincide, and let $R$ be its third intersection counted with <intersection multiplicity>. Draw the line through $R$ and $o$, again using the tangent if necessary, and let $S$ be its third intersection. Define $P+Q=S$. When $o$ is a flex, this is the familiar reflection of the first third-intersection point across the Weierstrass involution. The two-line construction also works when the chosen identity is not a flex.
To prove associativity without assuming it in the geometry, use <divisor class>. A nonsingular plane cubic has genus one. By the <Riemann-Roch theorem>, every <divisor class> of degree one contains exactly one effective <divisor on an algebraic curve> of degree one: for such a <divisor on an algebraic curve> $D$,
$$
\ell(D)-\ell(K_C-D)=\deg D=1,
$$
and $\deg(K_C-D)=-1$, so $\ell(K_C-D)=0$. Thus $\ell(D)=1$, and the nonzero section is unique up to scalar. Over $K$, a degree-one effective <divisor on an algebraic curve> defined over $K$ is a $K$-rational point. This also proves uniqueness of that point. In particular the map
$$
\iota:C(K)\longrightarrow\operatorname{Pic}^0(C),\qquad
P\longmapsto[P-o]
$$
is injective and identifies these points with the corresponding <degree-zero Picard group of a curve>: to recover the point from a class, add $[o]$ and take its unique effective representative. This is the <elliptic curve group law from Riemann-Roch>.
Let $H$ denote the <divisor class> cut out by a line. The two lines in the <chord-and-tangent group law> give
$$
P+Q+R\sim H,\qquad R+o+S\sim H.
$$
Their difference gives $P+Q\sim o+S$, hence
$$
\iota(S)=[P-o]+[Q-o].
$$
Addition of <divisor class> is associative. Therefore both $(P+Q)+R$ and $P+(Q+R)$ map to $[P-o]+[Q-o]+[R-o]$. Injectivity of $\iota$ proves \b[the <chord-and-tangent group law> is associative]. The same argument gives identity $o$, commutativity and inverses, and covers tangencies through the use of <intersection multiplicities>. All constructions are defined over $K$.
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