= Solution
The <hyperbolic lines> $AB$ and $CD$ are both perpendicular to $BC$. They are therefore <ultraparallel hyperbolic lines>, and $BC$ is their unique <common perpendicular of ultraparallel hyperbolic lines>.
The common perpendicular realizes the minimum <hyperbolic distance> between two ultraparallel lines. This follows directly in the <Poincaré half-plane model> after a <Möbius transformation> sends the two lines to concentric semicircles: the supplied distance formula is minimized along their common radial geodesic. The segment $AD$ also joins the two lines, but it is different from $BC$ in a nondegenerate quadrilateral. Hence
$$
\boxed{\rho(A,D)>\rho(B,C).}
$$
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