# Golden Ratio by Compass Only

In a 2002 article, K. Hofstetter, offered an elegant way of obtaining the Golden Ratio.

It will be convenient to denote S(R) the circle with center S through point R. For the construction, let A and B be two points. Circles A(B) and B(A) intersect in C and D and cross the line AB in points E and F. Circles E(B) and F(A) intersect in X and Y, as in the diagram. Because of the symmetry, points X, D, C, Y are collinear. The fact is

Assume for simplicity that AB = 2. Then

CX / CD | = (√15 + √3) / 2√3 |

= (√5 + 1) / 2 | |

= φ. |

Also, CD / DX = φ. Finally, observe that points E and F lie on C(D). It follows that the whole construction can be accomplished with compass only.

### References

- K. Hofstetter,
__A Simple Construction of the Golden Section__,*Forum Geometricorum*, v 2 (2002), pp. 65-66

### Fibonacci Numbers

- Ceva's Theorem: A Matter of Appreciation
- When the Counting Gets Tough, the Tough Count on Mathematics
- I. Sharygin's Problem of Criminal Ministers
- Single Pile Games
- Take-Away Games
- Number 8 Is Interesting
- Curry's Paradox
- A Problem in Checker-Jumping
- Fibonacci's Quickies
- Fibonacci Numbers in Equilateral Triangle
- Binet's Formula by Inducion
- Binet's Formula via Generating Functions
- Generating Functions from Recurrences
- Cassini's Identity
- Fibonacci Idendtities with Matrices
- GCD of Fibonacci Numbers
- Binet's Formula with Cosines
- Lame's Theorem - First Application of Fibonacci Numbers

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

Copyright © 1996-2017 Alexander Bogomolny62068015 |