This study begins with figures formed by four pairwise intersecting lines on a plane, including convex quadrilaterals, concave quadrilaterals, and butterfly-shaped figures. It systematically explores the cyclic properties of the intersection points of the internal and external angle bisectors of quadrilaterals and cyclic quadrilaterals. From the perspective that “a cyclic quadrilateral” and “a tangential quadrilateral” are dual figures, the study further hypothesizes whether “the figure enclosed by the perpendicular bisectors of a tangential quadrilateral” might possess a circle tangent to all its sides. The results are surprising: not only do the intersection points of the internal and external angle bisectors of quadrilaterals and cyclic quadrilaterals reveal additional cyclic properties, but the figure formed by the intersection points of the perpendicular bisectors of a tangential quadrilateral also exhibits cyclic phenomena. Moreover, several new properties emerge, such as the collinearity of the centers of these circles.