Dear All My Friends,
 
In "A simple construction of the golden section", Forum
Geometricorum, 11 (2011) 53:
 
http://forumgeom.fau.edu/FG2011volume11/FG201105index.html
 
Jo Niemeyer offered a beautiful way of constructing the
Golden Ratio with three equal segments.
 
It is interesting that Jo Niemeyer's construction can be
proved using Kurt Hofstetter's construction in "A Simple Construction
of the Golden Section", Forum Geometricorum, 2 (2002) 65--66:
 
http://forumgeom.fau.edu/FG2002volume2/FG200208index.html
 
Detail as following:
 
We use diagram in Jo Niemeyer's FG paper. L is perpendicular
bisector of A1A2. The circle B3(A1) with diameter A2B2 with center B3 passing
A1 cuts L at C, D (C is the same side with B3 wrt A1A2). The circle D(A1)
centered at D passing B3, A2 cuts L at E (other than B3). The circle B3(E) pass
through A3 because A3B3=A2B2=B3E. The circle D(C) pass through A3 because
symmetries.
Four circles B3(A1), D(A1), B3(E), D(C) bound exactly Kurt
Hofstetter's construction in above paper.
 
Please see figure:
 
http://a8.sphotos.ak.fbcdn.net/hphotos-ak-snc7/391946_309199489102127_2971652636\
38883_1035338_385991644_n.jpg
 
Best regards,
Bui Quang Tuan


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