Golden Section in Two Equilateral Triangles

Quang Tuan Bui

November 6, 2011

ABC and AMN are two equilateral triangles where M is midpoint of side BC. Arc 60° centered at B passing through A, C intersects side MN by golden ratio.

For a proof, construct

• D as symmetry of A in M,
• E as symmetry of A in N,
• Q as symmetry of P in M.

By George Odom, NM/MQ = φ; therefore MN/MP = φ.

(There is another proof for the same construction.)

Fibonacci Numbers

1. Ceva's Theorem: A Matter of Appreciation
2. When the Counting Gets Tough, the Tough Count on Mathematics
3. I. Sharygin's Problem of Criminal Ministers
4. Single Pile Games
5. Take-Away Games
6. Number 8 Is Interesting
7. Curry's Paradox
8. A Problem in Checker-Jumping
   • Getting a Scout out of Desert
9. Fibonacci's Quickies
10. Fibonacci Numbers in Equilateral Triangle
11. Binet's Formula by Inducion
12. Binet's Formula via Generating Functions
13. Generating Functions from Recurrences
14. Cassini's Identity
15. Fibonacci Idendtities with Matrices
16. GCD of Fibonacci Numbers
17. Binet's Formula with Cosines
18. Lame's Theorem - First Application of Fibonacci Numbers

• Golden Ratio in Geometry

|Contact| |Front page| |Contents| |Geometry| |Store|

Copyright © 1996-2015 Alexander Bogomolny