  Four points ordered in the positive order on the unit circle determine the
vertices of a quadrilateral, which is considered either as a euclidean or as a
hyperbolic quadrilateral depending on whether the lines connecting the vertices
are euclidean or hyperbolic lines. In the case of hyperbolic lines, this type
of quadrilaterals are called ideal quadrilaterals. Our main result gives a
euclidean counterpart of an earlier result on the hyperbolic distances between
the opposite sides of ideal quadrilaterals. The proof is based on computations
involving hyperbolic geometry. We also found a new formula for the hyperbolic
midpoint of a hyperbolic geodesic segment in the unit disk. As an application
of some geometric properties, we provided a euclidean construction of the
symmetrization of random four points on the unit circle with respect to a
diameter which preserves the absolute cross ratio of quadruples.
