Butterfly theorem

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

Let "M" be the midpoint of a 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 $XX\text{'},$ and $XX",$ be dropped from the point $X,$ on the straight lines $AM,$ and $DM,$ respectively. Similarly, let $YY\text{'},$ and $YY",$ be dropped from the point $Y,$ perpendicular to the straight lines $BM,$ and $CM,$ respectively.Now, since :: $riangle\; MXX\text{'}\; sim\; riangle\; MYY",,$: $\{MX\; over\; MY\}\; =\; \{XX\text{'}\; over\; YY"\},$

:: $riangle\; MXX"\; sim\; riangle\; MYY\text{'},,$: $\{MX\; over\; MY\}\; =\; \{XX"\; over\; YY\text{'}\},$

:: $riangle\; AXX\text{'}\; sim\; riangle\; CYY\text{'},,$: $\{XX\text{'}\; over\; YY\text{'}\}\; =\; \{AX\; over\; CY\},$

:: $riangle\; DXX"\; sim\; riangle\; BYY",,$: $\{XX"\; over\; YY"\}\; =\; \{DX\; over\; BY\},$

From the preceding equations, it can be easily seen that

: $left(\{MX\; over\; MY\}\; ight)^2\; =\; \{XX\text{'}\; over\; YY"\; \}\; \{XX"\; over\; YY\text{'}\},$

: $\{\}\; =\; \{AX.DX\; over\; CY.BY\},$

: $\{\}\; =\; \{PX.QX\; over\; PY.QY\},$

: $\{\}\; =\; \{(PM-XM).(MQ+XM)\; over\; (PM+MY).(QM-MY)\},$

: $\{\}\; =\; \{\; (PM)^2\; -\; (MX)^2\; over\; (PM)^2\; -\; (MY)^2\},$

since $PM\; ,$ = $MQ\; ,$

Now,

:$\{\; (MX)^2\; over\; (MY)^2\}\; =\; \{(PM)^2\; -\; (MX)^2\; over\; (PM)^2\; -\; (MY)^2\}.$

So, it can be concluded that$MX\; =\; MY,\; ,$ or $M\; ,$ is the midpoint of $XY.\; ,$

External links

* [http://agutie.homestead.com/files/GeometryButterfly.html Butterfly theorem, animated proof] by Antonio Gutierrez from "Geometry Step by Step from the Land of the Incas"
* [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.
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