NL-NWO 1965 Problem 3

Given are the points $A$ and $B$ in the plane. If $x$ is a straight line is in that plane, and $x$ does not coincide with the perpendicular bisectror of $AB$ , then denote the number of points $C$ located at $x$ such that $\vartriangle ABC$ is isosceles, as the "weight of the line $x$ ". Prove that the weight of any line $x$ is at most $5$ and determine the set of points $P$ which has a line with weight $1$ , but none with weight $0$ .