It is given to prove that
Every circle that passes through a fixed point and with its center on a fixed straight line, must pass through another fixed point.
As far as I have interpreted the question, what I need to prove is that
Any two circles that have a point $P$ common and have their centers lying on line $m$, they must have another common point $D$.
But as soon as I take tangential circles, I can easily disprove the very thing to be proved. Is my interpretation correct and does the question need any modification?