let $\Delta$ be the triangle whose corners $A$, $B$, $C$ points in general position in Euclidean plane and, let $D$ be a fourth point inside $\Delta$.
Question:
what is known about the construction of $D$ with $$\|D-A\|+\|C-B\|\ =\ \|D-B\|+\|A-C\|\ =\ \|D-C\|+\|B-A\|$$ i.e. for which all matchings of the $K_4$ induced by $A$, $B$, $C$ and $D$ have equal weight?
Does that center already have a name?
This is known as the point(s) of equal detour. This is usually defined as the point(s) $D$ such that $$|DA|+|DB|-|AB|=|DA|+|DC|-|AC|=|DB|+|DC|-|BC|$$ but this is easily seen to be equivalent to your definition. The reason I wrote "point(s)" is that sometimes a triangle can have two points of equal detour. Precise conditions of when this happens, as well as geometric constructions of this point (such as being the centers of the Soddy circles) can be found at the Encyclopedia of Triangle Centers under X(176). Or the original article The isoperimetric point and the point(s) of equal detour in a triangle by Hajja and Yff.
