Correct. Good job.
I wonder if there is a proof of this that is not overly complex?
Edit:
Well, No. I just found the proof, and it's not beautiful for its simplicity.
You start with the Law of Sines, and 2 1/2 pages later you have a symmetrical expression for one side.
"Do not try this at home."
oh i am so sorry .....i mistook it for the eulers line....my bad...

Two consecutive numbers are given to A and B. when A is asked ‘do you know B’s number?’ A denies. when B is asked whether he knows A’s no., B also denies. at the very moment, A replies that he knows B’s number. what may be B’s number?

You're right about the bisector case.
But for the trisector case there is more than one point.
In fact there are three places where a trisector of one angle first intersects a trisector of one of the other angles.
And there is something special about those three points.

hey man can you tell me how you worked out that the angle between alpha and beta should be 120 degrees .i just acant understand..Thanks
I knew someone would ask that question
It was a little messy and maybe not in the most efficient way, but...
thanks!