Problem
Given point $P$, the circunference $\Gamma$ whose center is point $C$ and whose radius is $r$ and the length $D$, find out the line $t$, such that $P \in t$, $\Gamma \cap t = \{A, B\}$ and $d(A,s)+d(B,s)=D$. See fig.1.

Figure $1$
Solution (See fig. 2.)

Figure $2$
Draw the line $s'$ such that $s' \parallel s $, $d(s,s')= \frac{D}{2}$ and $s' \cap \Gamma \neq \emptyset$.
Draw the line segment $CP$ and mark its midpoint $M$ on it.
Draw the circunference $\Lambda$ whose center is $M$ and whose radius is $CM$.
Find out the point $N$, such that $\{N\}=s' \cap \Lambda$ and $N$ is inside the circle delimited by $\Gamma$.
Draw the line $t$, such that $N \in t$ and $P \in t$.
Find out the points $A$ and $B$, such that $\{A,B\}= t \cap \Gamma$.
Explanation
Note that $N$ is the midpoint of $AB$ and $\angle CNA = \angle CNP = \frac{\pi}{2}$.
As $N$ is midpoint of $AB$, we have:
$d(A,s)+d(B,s)= 2d(N,s) \Rightarrow$ $d(A,s)+d(B,s)= 2 \left(\frac{D}{2}\right) \Rightarrow$ $d(A,s)+d(B,s)= D$