= Solution
Spatial homogeneity says that the velocity of $B$ relative to $A$ depends only on their separation. Comparing both galaxies with $C$ gives
$$
\mathbf v(\mathbf r_B-\mathbf r_A)
=\mathbf v(\mathbf r_B-\mathbf r_C)-\mathbf v(\mathbf r_A-\mathbf r_C).
$$
Thus $\mathbf v(\mathbf x-\mathbf y)=\mathbf v(\mathbf x)-\mathbf v(\mathbf y)$, so continuity of the macroscopic velocity field turns this <Cauchy functional equation> into a linear map:
$$
v_i=H_{ij}r_j.
$$
Isotropy requires $RH=HR$ for every <rotation matrix> $R$. The only such rank-two tensor is a scalar multiple of the identity, and hence
$$
\boxed{\mathbf v=H\mathbf r.}
$$
This is the <Hubble law>. A comoving galaxy has physical position $\mathbf r(t)=a(t)\mathbf x$, so
$$
\mathbf v=\dot a\mathbf x=\frac{\dot a}{a}\mathbf r,
\qquad
\boxed{H_0=\left.\frac{\dot a}{a}\right|_{t=t_0}.}
$$
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