I'm sorry, but I have no clue what you are saying. Could you draw a picture, scan it, and upload it?
There is a circle o and his chord AB which is not his diameter. On circle o we selcet point P, various from points A and B. Points Q and R lies respectively on lines PA and PB where QP=QB and RP=RA. Point M is a center of section QR. Prove that all obtained in this way lines PM (equivalent different location of point P on circle o) have a common point.
I came to that we can build a circle on quadrangle ABQR, but I do not know what do next.