Given a triangle ABC, how can you find a point P on AC and a point Q on BC, such that AP=PQ=QB?

*N.B. A construction method is sought, and only compass and straightedge are allowed.*

I just made my solution public. This problem was posed to me when in
High School, and I remember that my solution (the one given here) was
way easier and shorter than the teacher's, which I think involved
inversions of the plane or something like that.