# Golden Ratio in Square

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

### Construction

In square $ABCD$ $M,N,P,Q$ are midpoints of the sides, as shown below; $F$ the center of the square. Then circle $(AP)$ on $AP$ as a diameter, cuts $NF$ (point $X$) in Golden Ratio. Chord $AX$ cuts $MF$ in Golden Ratio:

Then $\displaystyle\frac{FX}{NX}=\frac{MZ}{FZ}=\phi,$ the golden ratio.

### Proof

Assume, without loss of generality, that $AB=2.$ Then $AP=\sqrt{5}.$ Since $AP$ is a diameter of the circle, $E$ - the intersection of $AP$ and $NQ$ is its center, which shows that $EX=\frac{1}{2}\sqrt{5}$ and that $EF=\frac{1}{2}.$ It follows that $FX=\frac{1}{2}(\sqrt{5}-1)$ and $NX=1-FX=\frac{1}{2}(3-\sqrt{5}),$ with a consequence that $\displaystyle\frac{FX}{NX}=\frac{1+\sqrt{5}}{2}=\phi.$

Next, triangles $AQX$ and $ZFX$ are similar: $\displaystyle\frac{FZ}{AQ}=\frac{FX}{QX},$ implying

$\displaystyle FZ=\frac{FX}{FX+1}=\frac{\sqrt{5}-1}{\sqrt{5}+1}=NX.$

We thus also have $MZ=FX$ and $\displaystyle\frac{MZ}{FZ}=\frac{FX}{NX}=\phi.$

### 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

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