What does really mean in the angle bisector containing the origin?

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" ...... algorithm may be used to find the bisector of the angle containing the origin....." what I meant is that there are two lines. When they meet, they form an acute angle ( say K) and also an angle of ( π - k) . The angular region which contains the origin is desired. Even though, I could use the formula to solve simple problems using the algorithm, but I can't get a very intuitive image of what really is happening in the algorithm. Providing a physical and intuitive picture so that one ( atleast I) can understand it completely would be of great help.