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Butterfly theorem

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Butterfly theorem

The butterfly theorem is a classical result in Euclidean geometry, which can be stated as follows:

Contents

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.

Proof

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

M X M Y = X X ′ Y Y ′ , M X M Y = X X ″ Y Y ″ , X X ′ Y Y ″ = A X C Y , X X ″ Y Y ′ = D X B Y ,

From the preceding equations, it can be easily seen that

( M X M Y ) 2 = X X ′ Y Y ′ X X ″ Y Y ″ , = A X ⋅ D X C Y ⋅ B Y , = P X ⋅ Q X P Y ⋅ Q Y , = ( P M − X M ) ⋅ ( M Q + X M ) ( P M + M Y ) ⋅ ( Q M − M Y ) , = ( P M ) 2 − ( M X ) 2 ( P M ) 2 − ( M Y ) 2 ,

since PM = MQ.

Now,

( M X ) 2 ( M Y ) 2 = ( P M ) 2 − ( M X ) 2 ( P M ) 2 − ( M Y ) 2 .

So, it can be concluded that MX = MY, or M is the midpoint of XY.

Other proofs exist, including one using projective geometry.

History

Proving the butterfly theorem was posed as a problem by William Wallace in The Gentlemen'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 Gentlemen's Diary or Mathematical Repository.

References

Butterfly theorem Wikipedia
(Text) CC BY-SA


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