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

PLEASE NOTE: Due to an error in the original proof, the prime symbols ( ' ) and double prime symbols ( " ) do not correlate properly with the picture or the written theorem. When reading, please note that Y' should be substituted with Y", and vice versa. X' and X" are correct.

[ 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, thee Wolfram Demonstrations Project.
• Eric W. Weisstein, Butterfly Theorem at MathWorld.