Golden Ratio in Inscribed Equilateral Triangles

Tran Quang Hung has posted on the CutTheKnotMath facebook page a sighting of the Golden Ratio in inscribed equilateral triangles. He also supplied the proof.

Statement

Equilateral $\Delta DEF$ is inscribed into equilateral $\Delta ABC$ so that its extended midlines $MN, NP, MP$ pass through the vertices of $\Delta ABC, as shown:$

golden ratio by Tran Quang Hung, construction #7

Then, say, the vertices of the inner triangle divide the sides of the outer triangle in the Golden Ratio, e.g., $\displaystyle\frac{BF}{AF}=\phi.$

Proof

Thales' theorem applied repeatedly gives the following sequence of proportions:

$\displaystyle\frac{AF}{BF}=\frac{MN}{BN}=\frac{FN}{AM}=\frac{BF}{AB},$

which is defining property of the Golden Section.

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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