In Euclidean geometry, the butterfly theorem is a classical result which can be stated as follows:Johnson, Roger A., Advanced Euclidean Geometry, Dover Publ., 2007 (orig. 1929).
Let be the midpoint of a chord of a circle, through which two other chords and are drawn; and intersect chord at and correspondingly. Then is the midpoint of .
Proof
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.
From the preceding equations and the intersecting chords theorem, it can be seen that
since . Thus,
Cross-multiplying the latter equation and cancelling out common terms,
Thus, is the midpoint of .
Other proofs exist,Martin Celli, "A Proof of the Butterfly Theorem Using the Similarity Factor of the Two Wings", Forum Geometricorum 16, 2016, 337–338. http://forumgeom.fau.edu/FG2016volume16/FG201641.pdf including one using projective geometry., problem 8.
History Proving the butterfly theorem was posed as a problem by William Wallace in The Gentleman's Mathematical Companion (1803). Three solutions were published in 1804, and in 1805 Sir William Herschel posed the question again in a letter to Wallace. Reverend Thomas Scurr asked the same question again in 1814 in the Gentleman's Diary or Mathematical Repository.William Wallace's 1803 Statement of the Butterfly Theorem, cut-the-knot, retrieved 2015-05-07.
References
External links
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.