Circles touching sides and the circumcircle for inscribed quadrilaterals

In an inscribed quadrilateral, four circles touching the circumcircle and two neighboring sides have a radical center. The main result of the article is the following theorem. Theorem 1. Let ABCD be a quadrilateral inscribed to a circle Ω. If Ωa is the circle touching Ω and segments AB, AD, and circles Ωb, Ωc, Ωd defined similarly (i. e. circles touching Ω and two neighboring sides of ABCD), then Ωa, Ωb, Ωc, and Ωd have a radical center (that is a point having equal powers with respect to Ωa, Ωb, Ωc, and Ωd).