Angle trisection drawn through formula_25 to the semicircle, resulting in the intersection formula_28. The angle formula_48 generated by formula_49 and formula_50 is the angle formula_51 sought. Another means to trisect an arbitrary angle by a "small" step outside the Greek framework is via a ruler with two marks a set distance apart. The next construction is originally due to Archimedes, called a "Neusis construction", i.e., that uses tools other than an "un-marked" straightedge. The diagrams we use show this construction for an acute angle, but it indeed works for any angle up to 180 degrees. This requires three facts from geometry (at
Less-than sign is used to denote the beginning of a heredoc statement (where OUTPUT is an arbitrary named variable.) In Bash, «<word is used as a "here string", where word is expanded and supplied to the command on its standard input, similar to a heredoc. The less-than sign plus the equals sign (<=) is used for an approximation of the less-than-or-equal-to sign (≤). ASCII does not have a less-than-or-equal-to sign, but Unicode defines it at code point U+2264. In BASIC, Lisp-family languages, and C-family languages (including Java and C++), operator <= means "less than or equal to". In Sinclair BASIC it is