Butterfly theorem

For the "butterfly lemma" of group theory, see Zassenhaus lemma.

Butterfly theorem:

M is the midpoint of XY.

In Euclidean geometry, the butterfly theorem is a classical result 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.

Proof

Proof of Butterfly theorem

  Chord PQ with known midpoint M.

  Perpendiculars from Y.

  Perpendiculars from X.

  Chords intersecting PQ at X, Y, and M.

To be proved: MX = MY.

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.

$${\begin{aligned}\triangle MXX'&\sim \triangle MYY',\\[2pt]\therefore {{\overline {MX}} \over {\overline {MY}}}&={{\overline {XX'}} \over {\overline {YY'}}}.\\[8pt]\triangle MXX''&\sim \triangle MYY'',\\[2pt]\therefore {{\overline {MX}} \over {\overline {MY}}}&={{\overline {XX''}} \over {\overline {YY''}}}.\\[8pt]\triangle AXX'&\sim \triangle CYY'',\\[2pt]\therefore {{\overline {XX'}} \over {\overline {YY''}}}&={{\overline {AX}} \over {\overline {CY}}}.\\[8pt]\triangle DXX''&\sim \triangle BYY',\\[2pt]\therefore {{\overline {XX''}} \over {\overline {YY'}}}&={{\overline {DX}} \over {\overline {BY}}}.\end{aligned}}$$

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

$${\begin{aligned}\left({{\overline {MX}} \over {\overline {MY}}}\right)^{2}&={{\overline {XX'}} \over {\overline {YY'}}}\cdot {{\overline {XX''}} \over {\overline {YY''}}},\\[2pt]&={{\overline {AX}}\cdot {\overline {DX}} \over {\overline {CY}}\cdot {\overline {BY}}},\\[2pt]&={{\overline {PX}}\cdot {\overline {QX}} \over {\overline {PY}}\cdot {\overline {QY}}},\\[2pt]&={{\bigl (}{\overline {PM}}-{\overline {XM}}{\bigr )}\cdot {\bigl (}{\overline {MQ}}+{\overline {XM}}{\bigr )} \over {\bigl (}{\overline {PM}}+{\overline {MY}}{\bigr )}\cdot {\bigl (}{\overline {QM}}-{\overline {MY}}{\bigr )}},\\[2pt]&={{\overline {PM}}^{2}-{\overline {MX}}^{2} \over {\overline {PM}}^{2}-{\overline {MY}}^{2}},\end{aligned}}$$

since PM = MQ. Thus,

$${{\overline {MX}}^{2} \over {\overline {MY}}^{2}}={{\overline {PM}}^{2}-{\overline {MX}}^{2} \over {\overline {PM}}^{2}-{\overline {MY}}^{2}}.$$

Cross-multiplying the latter equation and cancelling out common terms,

$${\begin{aligned}{\overline {MX}}^{2}\cdot {\overline {PM}}^{2}-{\overline {MX}}^{2}\cdot {\overline {MY}}^{2}&={\overline {MY}}^{2}\cdot {\overline {PM}}^{2}-{\overline {MX}}^{2}\cdot {\overline {MY}}^{2},\\[4pt]\Rightarrow \quad {\overline {MX}}^{2}\cdot {\overline {PM}}^{2}&={\overline {MY}}^{2}\cdot {\overline {PM}}^{2},\\[4pt]\Rightarrow \quad {\overline {MX}}^{2}&={\overline {MY}}^{2},\\[4pt]\Rightarrow \quad {\overline {MX}}&={\overline {MY}}.\end{aligned}}$$

Thus, 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 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.

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.
• Weisstein, Eric W. "Butterfly Theorem". MathWorld.