# Golden Ratio in Two Squares

Tran Quang Hung has posted on the CutTheKnotMath facebook page a simple construction of the Golden Ratio in Two Squares.

### Construction

Square $MNPC$ has point $M$ on the diagonal $BD$ of square $ABCD,$ $Q$ the intersection of $MP$ and $CD,$ extended.

If $M$ is such that $2DQ=CD$ then $AN/BN=\phi.$

### Proof

The proof is by Leo Giugiuc and uses complex numbers.

We choose $A=-1+i,$ $B=1+i,$ $C=1-i,$ $D=-1-i$ so that $Q=-2-i$ and $M=k+ki,$ $k\in (-1 ,1).$

$CN\perp QM$ and $QM$ has the slope $\displaystyle\frac{k+1}{k+2},$ implying $CN$ is described by the equation

$\displaystyle y+1=-\frac{k+2}{k+1}(x-1).$

Since $N\in AB$ then $N=x+i;$ so solving for $x,$ $x=-\displaystyle\frac{k}{k+2}$ and then for $N=-\displaystyle\frac{k}{k+2}+i.$ $NMCP$ is a positively oriented square, therefore, $\displaystyle\frac{N-M}{C-M}=i$ such that

$-k(k+3)+(1-k)(k+2)i=(k+2)[(k+1)+(1-k)i]$

which simplifies to $k^2+3k+1=0.$ Since $k\in (-1 ,1),$ $k=-1/\phi^{2},$ $N=(1/\phi^{2} )/\phi +i=1/\phi^{3} +i=\sqrt{5}-2+i$ so that, finally, $NA/NB=\phi.$

### 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
- Flat Probabilities on a Sphere
- Golden Ratio in Square And Circles
- Golden Ratio in Square
- Golden Ratio in Two Squares
- Golden Ratio in Isosceles Triangle
- Golden Ratio in Circles
- Golden Ratio in Isosceles Triangle II
- Golden Ratio in Yin-Yang

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