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{{For|the "butterfly lemma" of group theory|Zassenhaus lemma}}

[[Image:Butterfly theorem.svg|right|245px|thumb]]

The '''butterfly theorem''' is a classical result in [[Euclidean geometry]], which can be stated as follows:

Let ''M'' be the [[midpoint]] of a [[Chord (geometry)|chord]] ''PQ'' of a [[circle]], through which two other chords ''AB'' and ''CD'' are drawn; ''AD'' and ''BC'' intersect chord ''PQ'' at ''X'' and ''Y'' correspondingly. Then ''M'' is the midpoint of ''XY''.

A formal proof of the theorem is as follows:
Let the [[perpendiculars]] <math>XX'\,</math> and <math>XX''\,</math> be dropped from the point <math>X\,</math> on the straight lines <math>AM\,</math> and <math>DM\,</math> respectively. Similarly, let <math>YY'\,</math> and <math>YY''\,</math> be dropped from the point <math>Y\,</math> perpendicular to the straight lines <math>BM\,</math> and <math>CM\,</math> respectively.
[[Image:Proof of Butterfly theorem.png|right|frame|{{center|Proof of Butterfly theorem}}]]
Now, 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, it can be easily seen that

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

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

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

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

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

since <math>PM \,</math> = <math>MQ \,</math>

Now,

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

So, it can be concluded that
<math>MX = MY, \,</math> or <math>M \,</math> is the midpoint of <math>XY. \,</math>

An alternate proof using projective geometry can be found in problem 8 of the link below.

http://www.imomath.com/index.php?options=628&lmm=0

==Bibliography==
H. S. M. Coxeter, S. L. Greitzer, Geometry Revisited, MAA, 1967.

==External links==
* [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 in geometry]]
[[Category:Articles containing proofs]]

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