Butterfly theorem
From Wikipedia, the free encyclopedia
For the "butterfly lemma" of group theory, see Zassenhaus lemma.
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 
and 
be dropped from the point 
on the straight lines 
and 
respectively. Similarly, let 
and 
be dropped from the point 
perpendicular to the straight lines 
and 
respectively.
Now, since
From the preceding equations, it can be easily seen that
since 
= 
Now,
So, it can be concluded that 
or 
is the midpoint of 
[edit] External links
- Butterfly theorem, animated proof by Antonio Gutierrez from "Geometry Step by Step from the Land of the Incas"
- The Butterfly Theorem at cut-the-knot
- A Better Butterfly Theorem at cut-the-knot
- Proof of Butterfly Theorem at PlanetMath
- The Butterfly Theorem by Jay Warendorff, The Wolfram Demonstrations Project.
- Eric W. Weisstein, Butterfly Theorem at MathWorld.
