Butterfly set

The butterfly sentence ( English Butterfly theorem ) is a theorem of elementary geometry , of a property of chordal treated and as a corollary of tendons set using the similarity teaching results. The name is explained by the similarity between a plan figure for this sentence and the sketchy representation of a butterfly .

Formulation of the sentence

Given a circle with a chord . be the center of this tendon. and are two more circular tendons that also go through . And should be on the same side of . and let be the intersections of the circular chords or with . ${\ displaystyle [ST]}$ ${\ displaystyle M}$ ${\ displaystyle [AB]}$ ${\ displaystyle [CD]}$ ${\ displaystyle M}$ ${\ displaystyle A}$ ${\ displaystyle C}$ ${\ displaystyle [ST]}$ ${\ displaystyle P}$ ${\ displaystyle Q}$ ${\ displaystyle [AD]}$ ${\ displaystyle [CB]}$ ${\ displaystyle [ST]}$

Then the following applies: is also the midpoint of the route ${\ displaystyle M}$ ${\ displaystyle [PQ]}$ .

literature

William I. Jacobson: The Butterfly Problem Extensions, Generalizations . Mathematics Magazine, Vol. 42, No. 1 (Jan. 1969), pp. 17-21 ( JSTOR 2688031 )
Rudolf Fritsch : Comments on the butterfly theorem . Contributions to mathematics lessons 2003, Franzbecker, 2003, pp. 221–224
Claudi Alsina, Roger B. Nelsen: Pearls of Mathematics: 20 geometric figures as starting points for mathematical exploratory trips . Springer, 2015, ISBN 9783662454619 , p. 125

Web links

Commons : Butterfly theorem - collection of images, videos and audio files

Eric W. Weisstein : Butterfly Theorem . In: MathWorld (English). ; see. also in WolframDemonstationsProject
An improvement to the butterfly set (cut-the-knot)
Generalization of the butterfly theorem to ellipses