Butterfly theorem

File:Butterfly1.svg

A formal proof of the theorem is as follows:

Let the perpendiculars {{math|XX′}} and {{math|XX″}} be dropped from the point {{math|X}} on the straight lines {{math|AM}} and {{math|DM}} respectively. Similarly, let {{math|YY′}} and {{math|YY″}} be dropped from the point {{math|Y}} perpendicular to the straight lines {{math|BM}} and {{math|CM}} respectively.

Since

:: $\backslash triangle\; MXX\text{'}\; \backslash sim\; \backslash triangle\; MYY\text{'},$

: $\{MX\; \backslash over\; MY\}\; =\; \{XX\text{'}\; \backslash over\; YY\text{'}\},$

:: $\backslash triangle\; MXX$ \sim \triangle MYY,

: $\{MX\; \backslash over\; MY\}\; =\; \{XX$ \over YY},

:: $\backslash triangle\; AXX\text{'}\; \backslash sim\; \backslash triangle\; CYY\text{'}\text{'},$

: $\{XX\text{'}\; \backslash over\; YY\text{'}\text{'}\}\; =\; \{AX\; \backslash over\; CY\},$

:: $\backslash triangle\; DXX\text{'}\text{'}\; \backslash sim\; \backslash triangle\; BYY\text{'},$

:$\{XX\text{'}\text{'}\; \backslash over\; YY\text{'}\}\; =\; \{DX\; \backslash over\; BY\}.$

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

: $\backslash left(\{MX\; \backslash over\; MY\}\backslash right)^2\; =\; \{XX\text{'}\; \backslash over\; YY\text{'}\; \}\; \{XX$ \over YY},

: $\{\}\; =\; \{AX\; \backslash cdot\; DX\; \backslash over\; CY\; \backslash cdot\; BY\},$

: $\{\}\; =\; \{PX\; \backslash cdot\; QX\; \backslash over\; PY\; \backslash cdot\; QY\},$

: $\{\}\; =\; \{(PM-XM)\; \backslash cdot\; (MQ+XM)\; \backslash over\; (PM+MY)\; \backslash cdot\; (QM-MY)\},$

: $\{\}\; =\; \{\; (PM)^2\; -\; (MX)^2\; \backslash over\; (PM)^2\; -\; (MY)^2\},$

since {{math|PM {{=}} MQ}}.

So,

:$\{\; (MX)^2\; \backslash over\; (MY)^2\}\; =\; \{(PM)^2\; -\; (MX)^2\; \backslash over\; (PM)^2\; -\; (MY)^2\}.$

Cross-multiplying in the latter equation,

:$\{(MX)^2\; \backslash cdot\; (PM)^2\; -\; (MX)^2\; \backslash cdot\; (MY)^2\}\; =\; \{(MY)^2\; \backslash cdot\; (PM)^2\; -\; (MX)^2\; \backslash cdot\; (MY)^2\}\; .$

Cancelling the common term

:$\{\; -(MX)^2\; \backslash cdot\; (MY)^2\}$

from both sides of the equation yields

:$\{(MX)^2\; \backslash cdot\; (PM)^2\}\; =\; \{(MY)^2\; \backslash cdot\; (PM)^2\},$

hence {{math|MX {{=}} MY}}, since MX, MY, and PM are all positive, real numbers.

Thus, {{math|M}} is the midpoint of {{math|XY}}.

Other proofs exist,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 including one using projective geometry.[http://www.imomath.com/index.php?options=628&lmm=0], problem 8.

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.[http://www.cut-the-knot.org/pythagoras/WallaceButterfly.shtml William Wallace's 1803 Statement of the Butterfly Theorem], cut-the-knot, retrieved 2015-05-07.

{{reflist}}

{{commonscat|Butterfly theorem}}

• [http://www.cut-the-knot.org/pythagoras/Butterfly.shtml The Butterfly Theorem] at cut-the-knot
• [http://www.cut-the-knot.org/pythagoras/BetterButterfly.shtml A Better Butterfly Theorem] at cut-the-knot
• [http://planetmath.org/?op=getobj&from=objects&id=3613 Proof of Butterfly Theorem] at PlanetMath
• [http://demonstrations.wolfram.com/TheButterflyTheorem/ The Butterfly Theorem] by Jay Warendorff, the Wolfram Demonstrations Project.
• {{MathWorld |title=Butterfly Theorem |urlname=ButterflyTheorem}}

Category:Euclidean plane geometry

Category:Theorems about circles

Category:Articles containing proofs