Golden Ratio in Two Squares, Or, Perhaps in Three
Source
The following problem by Spt Ân (Vietnam) has been posted at the Peru Geometrico facebook group.
Problem
I prefer to reformulate the original problem by embedding it into an older configuration of three squares
Given three squares $ABCD,$ $DCEF,$ $FEGH$ and the circumcircle $\omega=(DCEF).$ Let $J=\omega\cap AG$ and $K=EJ\cap FH.$
Prove that $\displaystyle \frac{FK}{KH}=\frac{\sqrt{5}+1}{2}=\varphi,$ the Golden Ratio.
Proof
Let's denote $\angle CAG=\alpha.$ Then $\angle AGB=45^{\circ}-\alpha$ and $\displaystyle \tan(45^{\circ}-\alpha)=\frac{1}{3}.$ Setting $\tan\alpha=x,$ we get,
$\displaystyle \frac{1}{3}=\frac{1-x}{1+x},$
from which $\displaystyle x=\frac{1}{2}.$ By angle chasing, $\displaystyle \angle FEK=\angle FEJ=45^{\circ}-\frac{\alpha}{2}.$ If $\displaystyle y=\tan\frac{\alpha}{2},$ then $\displaystyle \frac{1}{2}=x=\frac{2y}{1-y^2}.$ Thus, choosing the positive value, $\displaystyle y=-2+\sqrt{5}.$ Further,
$\displaystyle \tan\angle FEK=\frac{1-y}{1+y}=\frac{3-\sqrt{5}}{\sqrt{5}-1}=\frac{\sqrt{5}-1}{2}.$
The meaning of this is that $\displaystyle \frac{FK}{EF}=\frac{FK}{FH}=\frac{\sqrt{5}-1}{2},$ so that $\displaystyle \frac{FK}{KH}=\frac{\sqrt{5}+1}{2}=\varphi,$ the Golden Ratio.
Extra
As the picture below (and above) shows, there are additional occurrences of the Golden Ratio.
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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