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Four points ordered in the positive order on the unit circle determine thevertices of a quadrilateral, which is considered either as a euclidean or as ahyperbolic quadrilateral depending on whether the lines connecting the verticesare euclidean or hyperbolic lines. In the case of hyperbolic lines, this typeof quadrilaterals are called ideal quadrilaterals. Our main result gives aeuclidean counterpart of an earlier result on the hyperbolic distances betweenthe opposite sides of ideal quadrilaterals. The proof is based on computationsinvolving hyperbolic geometry. We also found a new formula for the hyperbolicmidpoint of a hyperbolic geodesic segment in the unit disk. As an applicationof some geometric properties, we provided a euclidean construction of thesymmetrization of random four points on the unit circle with respect to adiameter which preserves the absolute cross ratio of quadruples.

On cyclic quadrilaterals in euclidean and hyperbolic geometries