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?
> 
> 
> 



Triangle Center from Weighted Perfect Matchings
This is known as the [point(s) of equal detour](http://mathworld.wolfram.com/EqualDetourPoint.html). 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](http://mathworld.wolfram.com/SoddyCircles.html)) can be found at the [Encyclopedia of Triangle Centers](http://faculty.evansville.edu/ck6/encyclopedia/ETC.html) under X(176). Or the original article [The isoperimetric point and the point(s) of equal detour in a triangle](https://link.springer.com/article/10.1007/s00022-007-1906-y) by Hajja and Yff.


