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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|XX′}} and {{math|XX″}} be dropped from the point {{math|X}} on the straight lines {{math|AM}} and {{math|DM}} respectively. Similarly, let {{math|YY′}} and {{math|YY″}} 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, 337–338.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

{{reflist}}

External links



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- time: 8:24pm EST - Sun, Nov 18 2018
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