Butterfly theorem

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The butterfly theorem is a classical result in Euclidean geometry, which can be stated as follows:<ref name=Johnson>Johnson, Roger A., Advanced Euclidean Geometry, Dover Publ., 2007 (orig. 1929).</ref>Template:Rp

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

Proof

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

Since

<math> \triangle MXX' \sim \triangle MYY',</math>

<math> {MX \over MY} = {XX' \over YY'}, </math>

<math> \triangle MXX \sim \triangle MYY,</math>

<math> {MX \over MY} = {XX \over YY}, </math>

<math> \triangle AXX' \sim \triangle CYY,</math>

<math> {XX' \over YY} = {AX \over CY}, </math>

<math> \triangle DXX \sim \triangle BYY',</math>

<math> {XX \over YY'} = {DX \over BY}. </math>

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

<math> \left({MX \over MY}\right)^2 = {XX' \over YY' } {XX \over YY}, </math>

<math> {} = {AX \cdot DX \over CY \cdot BY}, </math>

<math> {} = {PX \cdot QX \over PY \cdot QY}, </math>

<math> {} = {(PM-XM) \cdot (MQ+XM) \over (PM+MY) \cdot (QM-MY)}, </math>

<math> {} = { (PM)^2 - (MX)^2 \over (PM)^2 - (MY)^2}, </math>

since Template:Math.

So,

<math> { (MX)^2 \over (MY)^2} = {(PM)^2 - (MX)^2 \over (PM)^2 - (MY)^2}. </math>

Cross-multiplying in the latter equation,

<math> {(MX)^2 \cdot (PM)^2 - (MX)^2 \cdot (MY)^2} = {(MY)^2 \cdot (PM)^2 - (MX)^2 \cdot (MY)^2} . </math>

Cancelling the common term

<math> { -(MX)^2 \cdot (MY)^2} </math>

from both sides of the equation yields

<math> {(MX)^2 \cdot (PM)^2} = {(MY)^2 \cdot (PM)^2}, </math>

hence Template:Math, since MX, MY, and PM are all positive, real numbers.

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

Other proofs exist,<ref>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</ref> including one using projective geometry.<ref>[1], problem 8.</ref>

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.<ref>William Wallace's 1803 Statement of the Butterfly Theorem, cut-the-knot, retrieved 2015-05-07.</ref>

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.
• {{#invoke:Template wrapper|{{#if:|list|wrap}}|_template=cite web

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