Butterfly theorem

Template:Center Template:Center

In Euclidean geometry, the butterfly theorem is a classical result which can be stated as follows:Template:Rp

Let Template:Mvar be the midpoint of a chord Template:Mvar of a circle, through which two other chords Template:Mvar and Template:Mvar are drawn; Template:Mvar and Template:Mvar intersect chord Template:Mvar at Template:Mvar and Template:Mvar correspondingly. Then Template:Mvar is the midpoint of Template:Mvar.

Proof

Template:Center Template:Legend-line Template:Legend-line Template:Legend-line Template:Legend-line To be proved: Template:Math.

A formal proof of the theorem is as follows: Let the perpendiculars Template:Mvar and Template:Mvar be dropped from the point Template:Mvar on the straight lines Template:Mvar and Template:Mvar respectively. Similarly, let Template:Mvar and Template:Mvar be dropped from the point Template:Mvar perpendicular to the straight lines Template:Mvar and Template:Mvar respectively.

$$\begin{array}{cc}\mathrm{△}MX{X}^{\prime }& \sim \mathrm{△}MY{Y}^{\prime },\\ \therefore \frac{\overline{MX}}{\overline{MY}}& =\frac{\overline{X{X}^{\prime }}}{\overline{Y{Y}^{\prime }}}.\\ \mathrm{△}MX{X}^{″}& \sim \mathrm{△}MY{Y}^{″},\\ \therefore \frac{\overline{MX}}{\overline{MY}}& =\frac{\overline{X{X}^{″}}}{\overline{Y{Y}^{″}}}.\\ \mathrm{△}AX{X}^{\prime }& \sim \mathrm{△}CY{Y}^{″},\\ \therefore \frac{\overline{X{X}^{\prime }}}{\overline{Y{Y}^{″}}}& =\frac{\overline{AX}}{\overline{CY}}.\\ \mathrm{△}DX{X}^{″}& \sim \mathrm{△}BY{Y}^{\prime },\\ \therefore \frac{\overline{X{X}^{″}}}{\overline{Y{Y}^{\prime }}}& =\frac{\overline{DX}}{\overline{BY}}.\end{array}$$

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

$$\begin{array}{cc}{\left(\frac{\overline{MX}}{\overline{MY}}\right)}^2& =\frac{\overline{X{X}^{\prime }}}{\overline{Y{Y}^{\prime }}}\cdot \frac{\overline{X{X}^{″}}}{\overline{Y{Y}^{″}}},\\ & =\frac{\overline{AX}\cdot \overline{DX}}{\overline{CY}\cdot \overline{BY}},\\ & =\frac{\overline{PX}\cdot \overline{QX}}{\overline{PY}\cdot \overline{QY}},\\ & =\frac{(\overline{PM}-\overline{XM})\cdot (\overline{MQ}+\overline{XM})}{(\overline{PM}+\overline{MY})\cdot (\overline{QM}-\overline{MY})},\\ & =\frac{{\overline{PM}}^2-{\overline{MX}}^2}{{\overline{PM}}^2-{\overline{MY}}^2},\end{array}$$

since Template:Math. Thus,

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

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

$$\begin{array}{cc}{\overline{MX}}^2\cdot {\overline{PM}}^2-{\overline{MX}}^2\cdot {\overline{MY}}^2& ={\overline{MY}}^2\cdot {\overline{PM}}^2-{\overline{MX}}^2\cdot {\overline{MY}}^2,\\ \Rightarrow {\overline{MX}}^2\cdot {\overline{PM}}^2& ={\overline{MY}}^2\cdot {\overline{PM}}^2,\\ \Rightarrow {\overline{MX}}^2& ={\overline{MY}}^2,\\ \Rightarrow \overline{MX}& =\overline{MY}.\end{array}$$

Thus, Template:Mvar is the midpoint of Template:Mvar.

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.
• Template:MathWorld