Geometric Construction

Question

"Given in position the points $A,B$ and $P$ and a segment $m$, draw through $P$ a line $r$ in such a way that $A$ and $B$ be in opposite sides of $r$ and that the sum of the distances from $A$ and $B$ to $r$ be $m$."
Some hint of how to do it? Some link with the construction?

Answer 1

Here comes a proof without words:

$$\phantom{\sum}$$