A quick presentation how to trisect an angle using ruler-and-compass method.
See
https://en.wikipedia.org/wiki/Angle_trisection
The method uses Intercept theorem, see
https://en.wikipedia.org/wiki/Intercept_theorem
I couldn't find any issue with the method, but perhaps there is, since Pierre Wantzel claims it's impossible (see above Angle trisection on Wiki for details).