The butterfly theorem is a classical result in

Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...

, which can be stated as follows:Johnson, Roger A., ''Advanced Euclidean Geometry'', Dover Publ., 2007 (orig. 1929). Let be the

midpoint In geometry, the midpoint is the middle point of a line segment. It is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment. Formula The midpoint of a segment in ''n''-dimen ...

of a chord of a

circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...

, through which two other chords and are drawn; and intersect chord at and correspondingly. Then is the midpoint of .

Proof

A formal proof of the theorem is as follows: Let the perpendiculars and be dropped from the point on the straight lines and respectively. Similarly, let and be dropped from the point perpendicular to the straight lines and respectively. Since ::

\triangle MXX' \sim \triangle MYY',

= ,

\triangle MXX'' \sim \triangle MYY'',

= ,

\triangle AXX' \sim \triangle CYY'',

= ,

\triangle DXX'' \sim \triangle BYY',

= .

From the preceding equations and the

intersecting chords theorem The intersecting chords theorem or just the chord theorem is a statement in elementary geometry that describes a relation of the four line segments created by two intersecting chords within a circle. It states that the products of the lengths o ...

, it can be seen that :

\left(\right)^2 =  ,

= ,

= ,

= ,

= ,

since . So :

= .

Cross-multiplying in the latter equation, :

=  .

Cancelling the common term :

from both sides of the resulting equation yields :

= ,

hence , since MX, MY, and PM are all positive, real numbers. Thus, is the midpoint of . Other proofs exist, including one using projective geometry.

History

Proving the butterfly theorem was posed as a problem by

William Wallace Sir William Wallace ( gd, Uilleam Uallas, ; Norman French: ; 23 August 1305) was a Scottish knight who became one of the main leaders during the First War of Scottish Independence. Along with Andrew Moray, Wallace defeated an English army ...

in ''The Gentlemen's Mathematical Companion'' (1803). Three solutions were published in 1804, and in 1805

Sir William Herschel Frederick William Herschel (; german: Friedrich Wilhelm Herschel; 15 November 1738 – 25 August 1822) was a German-born British astronomer and composer. He frequently collaborated with his younger sister and fellow astronomer Caroline H ...

posed the question again in a letter to Wallace. Rev. Thomas Scurr asked the same question again in 1814 in the ''Gentlemen's Diary or Mathematical Repository''.William Wallace's 1803 Statement of the Butterfly Theorem

cut-the-knot Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet-born Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and Math ...

, retrieved 2015-05-07.

References

External links

The Butterfly Theorem
at

A Better Butterfly Theorem
at

Proof of Butterfly Theorem
at

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The Butterfly Theorem
by Jay Warendorff, the

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. * {{MathWorld , title=Butterfly Theorem , urlname=ButterflyTheorem Euclidean plane geometry Theorems about circles Articles containing proofs