Golden Ratio in Two Equilateral Triangles

Here a most elegant construction of the Golden Ratio posted by Tran Quang Hung (with a proof) at the CutTheKnotMath facebook page.

Construction

Proof

Let $M$ be the point of tangency of $(O)$ with $AD.$

Triangles $OMD$ and $AOD$ are similar. It follows that $\displaystyle\frac{MD}{MO}=\frac{OD}{OA}=2$. Thus, $MD=2OM=EF.$ By a property of tangent, $DM^2=DF\cdot DE.$ From this, $EF^2=DF\cdot DE,$ or $\displaystyle\frac{FD}{FE}=\frac{EF}{ED},$ as required.

Golden Ratio

1. Golden Ratio in Geometry
2. Golden Ratio in an Irregular Pentagon
3. Golden Ratio in a Irregular Pentagon II
4. Inflection Points of Fourth Degree Polynomials
5. Wythoff's Nim
6. Inscribing a regular pentagon in a circle - and proving it
7. Cosine of 36 degrees
8. Continued Fractions
9. Golden Window
10. Golden Ratio and the Egyptian Triangle
11. Golden Ratio by Compass Only
12. Golden Ratio with a Rusty Compass
13. From Equilateral Triangle and Square to Golden Ratio
14. Golden Ratio and Midpoints
15. Golden Section in Two Equilateral Triangles
16. Golden Section in Two Equilateral Triangles, II
17. Golden Ratio is Irrational
18. Triangles with Sides in Geometric Progression
19. Golden Ratio in Hexagon
20. Golden Ratio in Equilateral Triangles
21. Golden Ratio in Square
22. Golden Ratio via van Obel's Theorem
23. Golden Ratio in Circle - in Droves
24. From 3 to Golden Ratio in Semicircle
25. Another Golden Ratio in Semicircle
26. Golden Ratio in Two Squares
27. Golden Ratio in Two Equilateral Triangles
28. Golden Ratio As a Mathematical Morsel
29. Golden Ratio in Inscribed Equilateral Triangles
30. Golden Ratio in a Rhombus
31. Golden Ratio in Five Steps
32. Between a Cross and a Square
33. Four Golden Circles
34. Golden Ratio in Mixtilinear Circles
35. Golden Ratio in Isosceles Right Triangle, Square, and Semicircle

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