PA + PB = k

Tracing out such a locus is a fairly standard thing that is done in Sketchpad by defining the sum as a segment, creating a point on the segment to partition the length in 2, creating a circle with each partition as a radius, and tracing out the intersections.

What if you have three points, A, B, and C, in the plane and then determine a fourth point P by measuring the distances to the original three and having the sum of those distances be a constant?

PA + PB + PC = k

Here, I have constructed the Fermat Point P[0] which is the point where the sum is a minimum. Then I traced out three points with the constant sum approximately 10, 12 and 14 units.

I am looking for a more elegant way to trace out the locus corresponding to any given constant sum of distances to the vertices. I would also like to know what the curve is called and whether there is a way to graph a relation with that shape.

I am calling on the vast readership and commenters to help!