About the midpoint of a chord of a circle, through which two other chords are drawn
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:[1]:p. 78
Let M be the midpoint of a chordPQ 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.
Proof
A formal proof of the theorem is as follows:
Let the perpendicularsXX′ 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.
hence MX = MY, since MX, MY, and PM are all positive, real numbers.
Thus, M is the midpoint of XY.
Other proofs exist,[2] including one using projective geometry.[3]
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. Rev. Thomas Scurr asked the same question again in 1814 in the Gentleman's Diary or Mathematical Repository.[4]
This article uses material from the Wikipedia article Butterfly_theorem, and is written by contributors.
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