# Golden Ratio With Two Unequal Circles And a Line II

The following two constructions of the Golden Ratio are due to Nguyen Thanh Dung. The second one appears a fall back to the famous construction by George Odom, but the semblance is only superficial, the two constructions are distinct.

In the diagram, $AB=BC=CD,\;$ $AO=DO,\;$ $BM=MO;\;$ $TX\perp AD,\;$ $M\in TX.$

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Then the ratio of the blue segment to a red one is Golden: $\displaystyle\frac{YZ}{TZ}=\varphi.$

For a proof, assume $AB=4.\;$ Then in $\Delta CMZ,\;$ $CM=3,\;$ $CZ=4\;$ so that $MZ=\sqrt{7}\;$ and $YZ=2\sqrt{7}.$

In $\Delta OMT,\;$ $MO=1,\;$ $OT=6;\;$ so that $MT=\sqrt{35}.\;$ $TZ=\sqrt{35}-\sqrt{7}.\;$ It follows that

$\displaystyle\frac{YZ}{TZ}=\frac{2\sqrt{7}}{\sqrt{35}-\sqrt{7}}=\frac{2}{\sqrt{5}-1}=\varphi.$

For a related construction, let $AE\;$ and $AF\;$ be tangent to circle $(C),\;$ with $E,F\;$ on circle $(O).\;$ The $\Delta AEF\;$ is equilateral, with the circumcircle $(O)\;$ and a mixtilinear circle $(C).\;$ This could have been a point of departure.

To see that $\Delta AEF\;$ is equilateral, consider $\Delta ACG,\;$ as shown below:

Since $AC=8\;$ and $\displaystyle CG=4=\frac{AC}{2},\;$ $\angle CAG=30^{\circ},\;$ so that $\angle EAF=60^{\circ}.$

### 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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