= Solution
Let $T=(0,0)$, the rational point of order two. Define
$$
\alpha:E(\mathbb Q)\longrightarrow\mathbb Q^\times/(\mathbb Q^\times)^2
$$
by
$$
\alpha(O)=1,
\qquad
\alpha(T)=b,
\qquad
\alpha(x,y)=x\quad(x\ne0),
$$
where the right sides denote square classes.
To prove multiplicativity, let a nonvertical line $y=mx+c$ meet $E$ in points with $x$-coordinates $x_1,x_2,x_3$. Substitution gives the monic cubic
$$
x^3+(a-m^2)x^2+(b-2mc)x-c^2,
$$
so <Vieta formulas> give $x_1x_2x_3=c^2$. If the first two intersection points are $P,Q$, the third is $-(P+Q)$ and has the same $x$-coordinate as $P+Q$. Therefore
$$
\alpha(P)\alpha(Q)\alpha(P+Q)=1
$$
in the square-class group. Tangencies follow by taking a repeated root, and the vertical and exceptional cases give the stated values at $O$ and $T$. Thus $\alpha$ is a <group homomorphism>.
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