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### Butterfly theorem

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Butterfly theorem
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{{For|the "butterfly lemma" of group theory|Zassenhaus lemma}}(File:Butterfly theorem.svg|upright=1.0|thumb|{{center|Butterfly theorem}})The butterfly theorem is a classical result in Euclidean geometry, which can be stated as follows:Johnson, Roger A., Advanced Euclidean Geometry, Dover Publ., 2007 (orig. 1929).{{rp|p. 78}}Let {{math|M}} be the midpoint of a chord {{math|PQ}} of a circle, through which two other chords {{math|AB}} and {{math|CD}} are drawn; {{math|AD}} and {{math|BC}} intersect chord {{math|PQ}} at {{math|X}} and {{math|Y}} correspondingly. Then {{math|M}} is the midpoint of {{math|XY}}.

## Proof

(File:Butterfly1.svg|thumb|upright=1.0|{{center|Proof of Butterfly theorem}})A formal proof of the theorem is as follows:Let the perpendiculars {{math|XXâ€²}} and {{math|XXâ€³}} be dropped from the point {{math|X}} on the straight lines {{math|AM}} and {{math|DM}} respectively. Similarly, let {{math|YYâ€²}} and {{math|YYâ€³}} be dropped from the point {{math|Y}} perpendicular to the straight lines {{math|BM}} and {{math|CM}} respectively.Now, since
triangle MXX' sim triangle MYY',
{MX over MY} = {XX' over YY'},
triangle MXX sim triangle MYY,
{MX over MY} = {XX over YY},
triangle AXX' sim triangle CYY'',
{XX' over YY''} = {AX over CY},
triangle DXX'' sim triangle BYY',
{XX'' over YY'} = {DX over BY},
From the preceding equations and intersecting chords theorem, it can be easily seen that
left({MX over MY}right)^2 = {XX' over YY' } {XX over YY},
{} = {AX cdot DX over CY cdot BY},
{} = {PX cdot QX over PY cdot QY},
{} = {(PM-XM) cdot (MQ+XM) over (PM+MY) cdot (QM-MY)},
{} = { (PM)^2 - (MX)^2 over (PM)^2 - (MY)^2},
since {{math|PM {{=}} MQ}}.Now,
{ (MX)^2 over (MY)^2} = {(PM)^2 - (MX)^2 over (PM)^2 - (MY)^2}.
So, it can be concluded that {{math|MX {{=}} MY}}, or {{math|M}} is the midpoint of {{math|XY}}.Other proofs exist,Martin Celli, "A Proof of the Butterfly Theorem Using the Similarity Factor of the Two Wings", Forum Geometricorum 16, 2016, 337â€“338.weblink including one using projective geometry.weblink, problem 8.

## History

Proving the butterfly theorem was posed as a problem by William Wallace in The Gentlemen's Mathematical Companion (1803). Three solutions were published in 1804, and in 1805 Sir William Herschel 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, retrieved 2015-05-07.

## References

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