BG-MO 1968 Problem 4

On the line $g$ we are given the segment $AB$ and a point $C$ not on $AB$ . Prove that on $g$ , there exists at least one pair of points $P,Q$ symmetrical with respect to $C$ , which divide the segment $AB$ internally and externally in the same ratios, i.e $$\frac{PA}{PB}=\frac{QA}{QB} (1)$$ If $A,B,P,Q$ are such points from the line $g$ satisfying $(1)$ , prove that the midpoint $C$ of the segment $PQ$ is the external point for the segment $AB$ .

K. Petrov