# Golden Ratio in Mixtilinear Circles

Tran Quang Hung has found the Golden Ratio among mixtilinear circles in equilateral triangle.

I am surprised no more. However, the novel manifestation of the universality of the Golden Ratio has warranted some investigation. It appears that even a single mixtilinear circle in equilateral triangle is conducive to proliferation of that universal constant. Here's what I found:

($F$ is the center of the circle. The significance of other points is clear from the diagram.)

$\displaystyle\frac{GP}{JP}=\frac{GJ}{IJ}=\frac{KO}{JO}=\frac{GK}{KQ}=\frac{KO}{KL}=\frac{GQ}{CG}=\frac{KQ}{BQ}=\phi.$

As a matter of fact, rather often, elegant results in mathematics are established by proofs that lack appeal, let alone elegance, and provide no enlightenment into the nature of the statement. For me, this was the case with all of the above ratios. I confess to having proved the third one and getting a notion of how to proceed from there. I recoil from doing that.

$\displaystyle\frac{KO}{JO}=\phi.$

I'll work with $\Delta FJN:$

Assume the side of $\Delta ABC$ is $1.$ Then $AD=\displaystyle\frac{2}{3}$ and $\displaystyle FJ=DF=\frac{2\sqrt{3}}{9}.$ $DN$ is one third of the altitude from $C,$ i.e., $DN =\displaystyle\frac{\sqrt{3}}{6}.$ Thus, $JN^{2}=FJ^{2}-FN^{2},$ or more explicitly

$\displaystyle JN^{2}=\bigg(\frac{2\sqrt{3}}{9}\bigg)^{2}-\bigg(\frac{2\sqrt{3}}{9}-\frac{\sqrt{3}}{6}\bigg)^{2}=\frac{5}{36},$

implying $\displaystyle JN=\frac{\sqrt{5}}{6}.$ Further, $\displaystyle DO=KO=\frac{1}{3}$ and $\displaystyle NO=\frac{1}{6}.$ It follows that

$\begin{align}\displaystyle \frac{KO}{JO}&=\frac{1/3}{\sqrt{5}/6-1/6}\\ &=\frac{2}{\sqrt{5}-1}\\ &=\frac{\sqrt{5}+1}{2}=\phi. \end{align}$

(A solution by Leo Giugiuc to the original problem of two mixtilinear circles can be found in a separate file.)

### 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

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