Label the points of a triangle $p$, $q$, $r$ in $\mathbb{R}^2$.
Fixing both $p$ and $q$, I need to find the locus of points $r$ such that $$|r-p| - |r-q| = K$$ for some real number $K > 0$
Letting $q = (-c, 0)$, $p = (c, 0)$, $r = (r_1, r_2)$ for some $c > 0$. Then the problem becomes $$\sqrt{(r_1 - c)^2 + r_2^2} - \sqrt{(r_1 + c)^2 + r_2^2} = K$$
By solving this the above should follow, but from here all roads I try are dead-ends. Is there a known solution to such a problem?
The locus of such points whose difference of distances to 2 fixed points $p$ and $q$ is a branch of a hyperbola with foci $p,q$.
As $K>0$, $|r-p| > |r-q|$ : therefore, it is the branch which is closest to focus $q$.
See this ; unfortunately no difference of treatment is made there between the two branches.