# Golden Ratio In Three Tangent Circles

### Tran Quang Hung's variant

Given three pairwise tangent circles $(A),\;$ $(B),\;$ and $(C),\;$ let $F,\;$ $D,\;$ $E\;$ be the points of tangency. Line $DF\;$ intersects circles $(B),\;$ $(C),\;$ and $(ABC)\;$ successively in points $G,\;$ $H,\;$ $F,\;$ $D,\;$ $I,\;$ $J.$

Then $\displaystyle\frac{FH}{GH}=\frac{DF}{FH}=\frac{DH}{DJ}=\varphi,\;$ the Golden Ratio.

Assume $L\;$ is the shared center of $(ABC)\;$ and $(DEF)\;$ and that $DE=1.\;$ Also let $K\;$ be the midpoint of $DF.\;$

Then $AB=2\;$ and the radius $R\;$ of $(ABC)\;$ can be found to be $\displaystyle R=2\frac{2}{3}\frac{\sqrt{3}}{2}=\frac{2\sqrt{3}}{3}.\;$ Note that $HL=R.\;$ Further, $\displaystyle KL=\frac{1}{3}\frac{\sqrt{3}}{2}=\frac{\sqrt{3}}{6}.$

The Pythagorean theorem, applied in $\Delta HKL,\;$ then gives

$\displaystyle HK=\sqrt{ \left(\frac{2\sqrt{3}}{3}\right)^2-\left(\frac{\sqrt{3}}{6}\right)^2 }=\frac{\sqrt{5}}{2}.$

Now, $GF=1\;$ and $DG=2\;$ because $\Delta BFG\;$ is equilateral. The rest is left as an exercise.

### Elliot McGucken's variant

**The Construction**: Begin with an equilateral triangle. Inscribe a circle inside of it. Connect the two upper points where the circle is tangent to the triangle (segment $DI)\;$ and find the midpoint of the segment $(M).\;$ Find the midpoint of the side of the newly defined smaller triangle atop $(L).\;$ Draw a line from that midpoint $(L)\;$ through the midpoint $(M)\;$ and on to the border of the circle.

The red and blue line segments, thusly defined, are in the ratio of $1.618\;$ to $1.\;$ Also, the ratio of segment $p\;$ to segment $q\;$ in the upper triangle is also the golden ratio $\varphi.$

### Overlap

### Acknowledgment

The first part has been posted by Tran Quang Hung at the CutTheKnotMath facebook page. The second part has been posted earlier by Elliot McGucken at his facebook page and elsewhere. Elliot also provided the overlap diagram where his diagram is embedded into that of Tran Quang Hung. I thank them both.

### 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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