# Golden Ratio in Hexagon

Tran Quang Hung has posted on the CutTheKnotMath facebook page a simple construction of the golden ratio in hexagon.

### Construction

Square $ABHG$ is constructed outside the hexagon $ABCDEF.$ Circle $(A, CH)$ with center at $A$ and radius $AH$ cuts $EF$ at $I$ in Golden Ratio:

### Proof

Assume, without loss of generality, that $AB=AF=1.$ Then $AI=AH=\sqrt{2}.$

Set $\alpha = \angle FAI,$ $\beta =\angle AIF.$ Apply the Law of Sines in $\Delta AIF:$

$\displaystyle\frac{AF}{\sin\beta}=\frac{FI}{\sin\alpha}=\frac{AI}{\sin 120^{\circ}}.$

This can be rewritten as

$\displaystyle\frac{1}{\sin\beta}=\frac{FI}{\sin\alpha}=\frac{\sqrt{2}}{\sqrt{3}/2}=\frac{2\sqrt{6}}{3}.$

Thus we can identify $\beta:$ $\displaystyle\sin\beta=\frac{\sqrt{6}}{4}.$ Next we compute $\displaystyle\cos\beta=\frac{\sqrt{10}}{4}.$ Now, observe that $\alpha +\beta =60^{\circ},$ so that

$\begin{align}\displaystyle \sin\alpha &= \sin (60^{\circ}-\beta)\\ &= \sin 60^{\circ}\cdot\cos\beta-\cos 60^{\circ}\cdot\sin\beta\\ &=\frac{\sqrt{3}}{2}\cdot\frac{\sqrt{10}}{4}-\frac{1}{2}\cdot\frac{2\sqrt{6}}{3}\\ &=\frac{\sqrt{6}}{8}(\sqrt{5}-1). \end{align}$

From here

$\displaystyle FI = \sin\alpha\cdot\frac{2\sqrt{6}}{3}=\frac{\sqrt{6}}{8}(\sqrt{5}-1)\cdot\frac{2\sqrt{6}}{3}=\frac{\sqrt{5}-1}{2}=\frac{1}{\phi}.$

It follows that $\displaystyle\frac{EF}{FI}=\frac{FI}{EI}=\phi.$

A proof using Stewart's Theorem has been suggested by Leo Giugiuc. Apply Stewart's theorem to $\Delta AEF$ and the cevian $AI:$

$AF^{2}\cdot EI+AE^{2}\cdot FI=AI^{2}\cdot EF+EI\cdot FI\cdot AF.$

Substituting $AF=EF=1,$ $AI=\sqrt{2},$ $AE=\sqrt{3},$FI=x,$ $EI=1-x$ gives

$(1-x)+3x=2+(1-x)x.$

This reduces to the quadratic equation $x^{2}+x-1=0,$ with the only positive solution $\displaystyle x=\frac{\sqrt{5}-1}{2},$ which gives $1-x=\displaystyle\frac{3-\sqrt{5}}{2},$ and, finally, $\displaystyle\frac{x}{1-x}=\phi.$

### Remark

Tran Quang Hung's construction that employs the diagonal of a square as a radius of a circle is akin to that of M. Bataille. Here's two constructions in the same diagram:

According to M. Bataille, $\displaystyle\frac{BJ}{BK}=\phi;$ according to Tran Quang Hung, $\displaystyle\frac{BL}{CL}=\frac{FI}{EI}=\phi.$

A 2011 article by Quang Tuan Bui covers both cases and leads to a whole family of Golden Ratios in a regular hexagon. Tran Quang Hung's construction, like that of M. Battaille, has been discovered independently.

### 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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