Given a chord P Q of a circle, draw any other two chords A B and C D passing through its midpoint. Call the points where A D and B C meet P Q X and Y. Then M is also the midpoint of X Y. There are a number of proofs of this theorem, including those by W. G. Horner, Johnson, and Coxeter (1987, pp. 78 and 144). The latter concise proof employs projective geometry. The following proof is given by Coxeter and Greitzer. In the figure at right, drop perpendiculars x_1 and y_1 from X and Y to A B, and x_2 and Y_2 from X and Y to C D.

butterfly catastrophe | butterfly curve | butterfly effect | butterfly function | butterfly graph | butterfly lemma | butterfly polyiamond | chord | circle | cyclic quadrilateral | midpoint | quadrilateral

moths and butterflies