View source for Butterfly theorem

{{short description|About the midpoint of a chord of a circle, through which two other chords are drawn}}
{{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:<ref name=Johnson>Johnson, Roger A., ''Advanced Euclidean Geometry'', Dover Publ., 2007 (orig. 1929).</ref>{{rp|p. 78}}

Let {{math|''M''}} be the [[midpoint]] of a [[Chord (geometry)|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.

Since 
:: <math> \triangle MXX' \sim \triangle MYY',</math>
: <math> {MX \over MY} = {XX' \over YY'}, </math>

:: <math> \triangle MXX'' \sim \triangle MYY'',</math>
: <math> {MX \over MY} = {XX'' \over YY''}, </math>

:: <math> \triangle AXX' \sim \triangle CYY'',</math>
: <math> {XX' \over YY''} = {AX \over CY}, </math>

:: <math> \triangle DXX'' \sim \triangle BYY',</math>
:<math> {XX'' \over YY'} = {DX \over BY}. </math>

From the preceding equations and the [[intersecting chords theorem]], it can be seen that

: <math> \left({MX \over MY}\right)^2 = {XX' \over YY' } {XX'' \over YY''}, </math>

: <math> {} = {AX \cdot DX \over CY \cdot BY}, </math>

: <math> {} = {PX \cdot QX \over PY \cdot QY}, </math>

: <math> {} = {(PM-XM) \cdot (MQ+XM) \over (PM+MY) \cdot (QM-MY)}, </math>

: <math> {} = { (PM)^2 - (MX)^2 \over (PM)^2 - (MY)^2}, </math>

since {{math|''PM'' {{=}} ''MQ''}}.

So,

:<math> { (MX)^2 \over (MY)^2} = {(PM)^2 - (MX)^2 \over (PM)^2 - (MY)^2}. </math>

Cross-multiplying in the latter equation,

:<math> {(MX)^2 \cdot (PM)^2 - (MX)^2 \cdot (MY)^2} = {(MY)^2 \cdot (PM)^2 - (MX)^2 \cdot (MY)^2} . </math>

Cancelling the common term

:<math> { -(MX)^2 \cdot (MY)^2} </math>

from both sides of the equation yields

:<math> {(MX)^2 \cdot (PM)^2} = {(MY)^2 \cdot (PM)^2}, </math>

hence {{math|''MX'' {{=}} ''MY''}}, since MX, MY, and PM are all positive, real numbers.

Thus, {{math|''M''}} is the midpoint of {{math|''XY''}}.

Other proofs exist,<ref>Martin Celli, "A Proof of the Butterfly Theorem Using the Similarity Factor of the Two Wings", ''[[Forum Geometricorum]]'' 16, 2016, 337–338. http://forumgeom.fau.edu/FG2016volume16/FG201641.pdf</ref> including one using [[projective geometry]].<ref>[http://www.imomath.com/index.php?options=628&lmm=0], problem 8.</ref>

==History==
Proving the butterfly theorem was posed as a problem by [[William Wallace (mathematician)|William Wallace]] in ''The Gentleman'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. Reverend Thomas Scurr asked the same question again in 1814 in the ''Gentleman's Diary or Mathematical Repository''.<ref>[http://www.cut-the-knot.org/pythagoras/WallaceButterfly.shtml William Wallace's 1803 Statement of the Butterfly Theorem], [[cut-the-knot]], retrieved 2015-05-07.</ref>

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==References==
{{reflist}}

==External links==
{{commonscat|Butterfly theorem}}
* [http://www.cut-the-knot.org/pythagoras/Butterfly.shtml The Butterfly Theorem] at [[cut-the-knot]]
* [http://www.cut-the-knot.org/pythagoras/BetterButterfly.shtml A Better Butterfly Theorem] at [[cut-the-knot]]
* [http://planetmath.org/?op=getobj&from=objects&id=3613 Proof of Butterfly Theorem] at [[PlanetMath]]
* [http://demonstrations.wolfram.com/TheButterflyTheorem/ The Butterfly Theorem]  by Jay Warendorff, the [[Wolfram Demonstrations Project]].
* {{MathWorld |title=Butterfly Theorem |urlname=ButterflyTheorem}}

[[Category:Euclidean plane geometry]]
[[Category:Theorems about circles]]
[[Category:Articles containing proofs]]