# Rotating Square in Search of the Golden Ratio

### John Arioni

18 March, 2017

Draw square $ABCD\,$ with sides equal to $1\,$ (side $AB\,$ horizontal) and rotate it counterclockwise around vertex $D\,$ through angle $\displaystyle \frac{\pi}{6}.\,$ Then draw a circle $A(C)\,$ with cener $A\,$ through $C.\,$ Obviously, the radius of the circle equals $\sqrt{2}.\,$ Let $E\,$ be on the circle, making $DE\,$ horizontal, as in the diagran below:

Note that $\angle ADE=\displaystyle \frac{\pi}{3}.\,$ Apply the Law of Cosines in $\Delta ADE:$

$\displaystyle AE^2=AD^2+DE^2-2AD\cdot DE\cos\frac{\pi}{3},$

so, letting x=DE, we have

$\displaystyle x^2-x+(1-2)=x^2-x-1=0.$

Taking the positive root we find that $x=\varphi,\,$ the Golden Ratio. Let $F\,$ be the second intersection of $DE\,$ with the circle:

We shall prove that $DF=\displaystyle \frac{1}{\varphi}.\,$ To this end consider the simplified diagram:

We have $AH=\sin\displaystyle \frac{\pi}{3}=\frac{3}{2},\,$ so that $HF=\sqrt{AF^2-AH^2}=\displaystyle \frac{\sqrt{5}}{2};\,$ $HD=\cos\displaystyle \frac{\pi}{3}=\frac{1}{2},\,$ and, finally, $DF=HF-HD=\displaystyle \frac{\sqrt{5}}{2}-\frac{1}{2}=\frac{1}{\varphi}.$

Obviously, if $G\,$ is the former (before the rotation) position of point $C\,$ then $\displaystyle \frac{EG}{ED}=\frac{ED}{DG}=\varphi:$

### Golden Ratio

- Golden Ratio in Geometry
- Golden Ratio in Regular Pentagon
- Golden Ratio in an Irregular Pentagon
- Golden Ratio in a Irregular Pentagon II
- Inflection Points of Fourth Degree Polynomials
- Wythoff's Nim
- Inscribing a regular pentagon in a circle - and proving it
- Cosine of 36 degrees
- Continued Fractions
- Golden Window
- Golden Ratio and the Egyptian Triangle
- Golden Ratio by Compass Only
- Golden Ratio with a Rusty Compass
- From Equilateral Triangle and Square to Golden Ratio
- Golden Ratio and Midpoints
- Golden Section in Two Equilateral Triangles
- Golden Section in Two Equilateral Triangles, II
- Golden Ratio is Irrational
- Triangles with Sides in Geometric Progression
- Golden Ratio in Hexagon
- Golden Ratio in Equilateral Triangles
- Golden Ratio in Square
- Golden Ratio via van Obel's Theorem
- Golden Ratio in Circle - in Droves
- From 3 to Golden Ratio in Semicircle
- Another Golden Ratio in Semicircle
- Golden Ratio in Two Squares
- Golden Ratio in Two Equilateral Triangles
- Golden Ratio As a Mathematical Morsel
- Golden Ratio in Inscribed Equilateral Triangles
- Golden Ratio in a Rhombus
- Golden Ratio in Five Steps
- Between a Cross and a Square
- Four Golden Circles
- Golden Ratio in Mixtilinear Circles
- Golden Ratio With Two Equal Circles And a Line
- Golden Ratio in a Chain of Polygons, So to Speak
- Golden Ratio With Two Unequal Circles And a Line
- Golden Ratio In a 3x3 Square
- Golden Ratio In a 3x3 Square II
- Golden Ratio In Three Tangent Circles
- Golden Ratio In Right Isosceles Triangle
- Golden Ratio Poster
- Golden Ratio Next to the Poster
- Golden Ratio In Rectangles
- Golden Ratio In a 2x2 Square: Without And Within
- Golden Ratio With Two Unequal Circles And a Line II
- Golden Ratio in Equilateral and Right Isosceles Triangles
- Golden Ratio in a Butterfly Astride an Equilateral Triangle
- The Golden Pentacross
- 5-Step Construction of the Golden Ratio, One of Many
- Golden Ratio in 5-gon and 6-gon
- Golden Ratio in an Isosceles Trapezoid with a 60 degrees Angle
- Golden Ratio in Pentagon And Two Squares
- Golden Ratio in Pentagon And Three Triangles
- Golden Ratio in a Mutually Beneficial Relationship
- Star, Six Pentagons and Golden Ratio
- Rotating Square in Search of the Golden Ratio
- Cultivating Regular Pentagons
- Golden Ratio in an Isosceles Trapezoid with a 60 degrees Angle II
- More of Gloden Ratio in Equilateral Triangles
- Golden Ratio in Three Regular Pentagons
- Golden Ratio in Three Regular Pentagons II
- Golden Ratio in Wu Xing
- Golden Ratio In Three Circles And Common Secant
- Flat Probabilities on a Sphere
- Golden Ratio in Square And Circles
- Golden Ratio in Square
- Golden Ratio in Two Squares, Or, Perhaps in Three
- Golden Ratio in Isosceles Triangle
- Golden Ratio in Circles
- Golden Ratio in Isosceles Triangle II
- Golden Ratio in Yin-Yang

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