MATHEMAGIC 3: ANGLES

Table with exact goniometric values (3 pages - special thanks to Chris Wildhagen)

A proof without words that  arctan(1) + arctan (2) + arctan (3) = π.
Can you see it?

Can you find out what is the correct value of tan (22.5°)?

My FIVE DISCS THEOREM.

Place 5 discs with radius r in a symmetric way on a larger disc with radius R (as indicated on the picture) so that these five discs cover the larger disc completely. If r is the smallest possible radius to do this, then the ratio R/r equals the golden ratio.

 

A proof (in Dutch) is given on
http://blogimages.bloggen.be/gnomon/attach/111768.pdf