A circle cuts out equal chords on all sides of the quadrilateral. Prove that it is possible to inscribe a circle in that quadrilateral.
Let O be the center of given circle, R – its radius, a – length of the chords cut out by the circle on quadrilateral’s sides. Then the distances from O to the sides of the quadrilateral are (R2-a2)/4, i.e. it’s equidistant from quadrilateral’s sides and thus is a center of inscribed circle.