To get a flavour of how this sort of problem might be solved, let's start by dividing a given angle into two equal parts, rather than three.
Bisecting an angle
Given two lines that meet at a point $O,$ you can bisect the angle between them as follows. Put the point of the compass at $O$ and draw a circle (of any radius you like). This is the blue circle in the diagram. The circle will cross the two lines at two points: call these $A$ and $B$.