The point inside a triangle where $a+f=c+e=b+d$ if $a, b$ and $c$ are side lengths and $d, e$ and $f$ are distances to the corners

Question

I am trying to find the general formula for finding the coordinate of a point within a triangle satisfying the condition $a+f = c+e = b+d$ where $a, b$ and $c$ are the lengths of the sides of the triangle and $d, e$ and $f$ are the distances between the searched point and the corners of the triangle as shown in the figure.

The coordinates of the corners of the triangle are known and will be used for finding the coordinate of the point of interest.

I am sorry if this is a simple problem for which the answer is known and the searched point has a special name, but I could not find an online explanation.

Thanks.

The image shows an example triangle but the answer is searched for all types of triangles.

Answer 1

This point it's an intersection of two hyperbolas: $$e-f=a-c$$ with focuses $A$ and $C$ and $$d-f=a-b$$ with focuses $A$ and $B$.