Butterfly Theorem


Given a chord PQ of a circle, draw any other two chords AB and CD passing through its midpoint. Call the points where AD and BC meet PQ X and Y. Then M is also the midpoint of XY. There are a number of proofs of this theorem, including those by W. G. Horner, Johnson (1929, p. 78), and Coxeter (1987, pp. 78 and 144). The latter concise proof employs projective geometry.

The following proof is given by Coxeter and Greitzer (1967, p. 46). In the figure at right, drop perpendiculars x_1 and y_1 from X and Y to AB, and x_2 and Y_2 from X and Y to CD. Write a=PM=MQ, x=XM, and y=MY, and then note that by similar triangles




so x=y. Q.E.D.

See also

Butterfly Catastrophe, Butterfly Curve, Butterfly Effect, Butterfly Function, Butterfly Graph, Butterfly Lemma, Butterfly Polyiamond, Chord, Circle, Cyclic Quadrilateral, Midpoint, Quadrilateral

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Bogomolny, A. "The Butterfly Theorem.", A. "A Better Butterfly Theorem.", A. "Two Butterflies Theorem.", H. S. M. Projective Geometry, 2nd ed. New York: Springer-Verlag, pp. 78 and 144, 1987.Coxeter, H. S. M. and Greitzer, S. L. "The Butterfly." §2.8 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 45-46, 1967.Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, p. 78, 1929.

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Butterfly Theorem

Cite this as:

Weisstein, Eric W. "Butterfly Theorem." From MathWorld--A Wolfram Web Resource.

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