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We divide any angle by this method into the ratio of x:y:x but not x:x:x so the trisection method you proposed is actually a method for partitioning an angle into ratio x:y:x (where values of x and y can be calculated for general case. You can try it and show that x is not equal to y !!!). Talking generally, we can divide any arbitrary angle into a paliandromic ratio by this chord sectioned method.(i.e. for 5 parts, angle wil be divided into ratio a:b:c:b:a , etc). There are different auxilliary curves like hyperbola(for trisection only), spiral(for n section), sinusoidal wave(for n section), etc which can trisect or nsect any given angle. Sometimes only intuition can give wrong results bro so read more and derive more.