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

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

:: $$riangle AXX' sim riangle CYY',, : $$ {XX' over YY'} = {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' over YY" } {XX" over YY'},

: $$ {} = {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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