Butterfly Theorem Geometric Construction

This GeoGebra applet demonstrates the classic Butterfly Theorem in Euclidean geometry. It constructs a circle with a chord PQ and its midpoint M. Two additional chords AB and CD are drawn through M. The intersections of AD and BC with PQ yield points X and Y, proving that M is also the midpoint of XY. The construction utilizes Circle, Point, Segment, Midpoint, Line, and Intersect commands to visually present the symmetric properties of the butterfly-shaped quadrilateral and the midpoint bisection relationship. Students can interactively drag points on the circle to observe how the configuration dynamically preserves the property that M remains the midpoint of XY regardless of the positions of the chords. This hands-on exploration reinforces understanding of circle geometry, chord properties, midpoint theorems, and the elegance of classical geometric proofs. The applet is well suited for secondary school and undergraduate geometry courses, serving as both a visual proof aid and an interactive exploration tool that deepens conceptual intuition about symmetry and bisection in circular figures.