Ruler and Compass Construction 19
Golden Rectangle - sides in golden ratio (1+sqrt(5))/2

Reset, segment AM of arbitrary length,
taken as unit length.
1. Extend ray A
2. Point B with MB = AM
3. Semi-circle centered at B, through A
4. Arc centered at A, through C
4. Arc centered at C, through A
5. Perpendicular BD
6. Intersection point E
7. Segment MF (length sqrt(5))
8. Arc centered at M through E
9. Arc centered at F, radius AB
10. Arc centered at E, radius BF
11. Arc centered at E, radius AB
12. Arc centered at A, through B
13. Golden rectangle AFGH

Why it works:
Treating AM as unit length, then MB = 1 and BE = 2, so ME = sqrt(5) and AF = 1 + sqrt(5). Thus
the ratio AF:EB is (1+sqrt(5))/2, the Golden Ratio.