Butterfly theorem

About the midpoint of a chord of a circle, through which two other chords are drawn
Butterfly theorem

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 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

Proof of Butterfly theorem

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.

Since

△ M X X ′ ∼ △ M Y Y ′ , {\displaystyle \triangle MXX'\sim \triangle MYY',}
M X M Y = X X ′ Y Y ′ , {\displaystyle {MX \over MY}={XX' \over YY'},}
△ M X X ″ ∼ △ M Y Y ″ , {\displaystyle \triangle MXX''\sim \triangle MYY'',}
M X M Y = X X ″ Y Y ″ , {\displaystyle {MX \over MY}={XX'' \over YY''},}
△ A X X ′ ∼ △ C Y Y ″ , {\displaystyle \triangle AXX'\sim \triangle CYY'',}
X X ′ Y Y ″ = A X C Y , {\displaystyle {XX' \over YY''}={AX \over CY},}
△ D X X ″ ∼ △ B Y Y ′ , {\displaystyle \triangle DXX''\sim \triangle BYY',}
X X ″ Y Y ′ = D X B Y . {\displaystyle {XX'' \over YY'}={DX \over BY}.}

From the preceding equations and the intersecting chords theorem, it can be seen that

( M X M Y ) 2 = X X ′ Y Y ′ X X ″ Y Y ″ , {\displaystyle \left({MX \over MY}\right)^{2}={XX' \over YY'}{XX'' \over YY''},}
= A X ⋅ D X C Y ⋅ B Y , {\displaystyle {}={AX\cdot DX \over CY\cdot BY},}
= P X ⋅ Q X P Y ⋅ Q Y , {\displaystyle {}={PX\cdot QX \over PY\cdot QY},}
= ( P M − X M ) ⋅ ( M Q + X M ) ( P M + M Y ) ⋅ ( Q M − M Y ) , {\displaystyle {}={(PM-XM)\cdot (MQ+XM) \over (PM+MY)\cdot (QM-MY)},}
= ( P M ) 2 − ( M X ) 2 ( P M ) 2 − ( M Y ) 2 , {\displaystyle {}={(PM)^{2}-(MX)^{2} \over (PM)^{2}-(MY)^{2}},}

since PM = MQ.

So,

( M X ) 2 ( M Y ) 2 = ( P M ) 2 − ( M X ) 2 ( P M ) 2 − ( M Y ) 2 . {\displaystyle {(MX)^{2} \over (MY)^{2}}={(PM)^{2}-(MX)^{2} \over (PM)^{2}-(MY)^{2}}.}

Cross-multiplying in the latter equation,

( M X ) 2 ⋅ ( P M ) 2 − ( M X ) 2 ⋅ ( M Y ) 2 = ( M Y ) 2 ⋅ ( P M ) 2 − ( M X ) 2 ⋅ ( M Y ) 2 . {\displaystyle {(MX)^{2}\cdot (PM)^{2}-(MX)^{2}\cdot (MY)^{2}}={(MY)^{2}\cdot (PM)^{2}-(MX)^{2}\cdot (MY)^{2}}.}

Cancelling the common term

− ( M X ) 2 ⋅ ( M Y ) 2 {\displaystyle {-(MX)^{2}\cdot (MY)^{2}}}

from both sides of the resulting equation yields

( M X ) 2 ⋅ ( P M ) 2 = ( M Y ) 2 ⋅ ( P M ) 2 , {\displaystyle {(MX)^{2}\cdot (PM)^{2}}={(MY)^{2}\cdot (PM)^{2}},}

hence MX = MY, since MX, MY, and PM are all positive, real numbers.

Thus, M is the midpoint of XY.

Other proofs too 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]


References

  1. ^ Johnson, Roger A., Advanced Euclidean Geometry, Dover Publ., 2007 (orig. 1929).
  2. ^ 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
  3. ^ [1], problem 8.
  4. ^ William Wallace's 1803 Statement of the Butterfly Theorem, cut-the-knot, retrieved 2015-05-07.
Wikimedia Commons has media related to Butterfly theorem.
  • 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.
  • Weisstein, Eric W. "Butterfly Theorem". MathWorld.