# Golden Ratio With Two Unequal Circles And a Line

The following invention of Tran Quang Hung, was likely inspired by an earlier construction of Elliot McGucken.

Given circle $(O)\;$ diameter $AB.\;$ Construct the circle $(K)\;$ diameter $OB.\;$ Let $C\;$ be the midpoint of arc $AB;\;$ $M,N\;$ the intersections of $CK\;$ with $(K).\;$

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Then $M\;$ divides $CN\;$ in the Golden Ratio: $\displaystyle\frac{MN}{CM}=\varphi.$

The proof is as simple as the construction. Assume $OK=KM=KN=1.\;$ Then $CK=\sqrt{5},\;$ $MN=2,\;$ $CM=\sqrt{5}-1;\;$ $\displaystyle\frac{MN}{CM}=\frac{2}{\sqrt{5}-1}=\frac{\sqrt{5}+1}{2}.$

Xuan Thang Vu offered a different approach: $CO,\;$ being a tangent to $(K),\;$ $CO^2=CM\cdot CN,\;$ or, say, $a^2=x(x+a).\;$ With $\displaystyle y=\frac{a}{x},\;$ $y^2=y+1,\;$ making $y=\varphi.$

Tran Quang Hung has observed additional occurrencies of the Golden Ratio that could be surmised in the derived diagram:

### 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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