Euclidean circle representing a perpendicular hyperbolic line

If $P$ is at hyperbolic distance $\rho$ from the origin, the Poincare-disc geodesic through $P$ perpendicular to $OP$ is represented by a Euclidean circle of radius $\mathrm{csch}\rho$ whose centre is at Euclidean distance $\coth \rho$ from the origin.