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# Butterfly Theorem

Given a chord of a circle, draw any other two chords and passing through its midpoint. Call the points where and meet and . Then is also the midpoint of . 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 and from and to , and and from and to . Write , , and , and then note that by similar triangles

 (1)
 (2)
 (3)

so

 (4) (5)

so . Q.E.D.

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

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## References

Bogomolny, A. "The Butterfly Theorem." http://www.cut-the-knot.org/pythagoras/Butterfly.shtml.Bogomolny, A. "A Better Butterfly Theorem." http://www.cut-the-knot.org/pythagoras/BetterButterfly.shtml.Bogomolny, A. "Two Butterflies Theorem." http://www.cut-the-knot.org/pythagoras/BetterButterfly.shtml.Coxeter, 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.

## Referenced on Wolfram|Alpha

Butterfly Theorem

## Cite this as:

Weisstein, Eric W. "Butterfly Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ButterflyTheorem.html