How do I Construct these Loci?

If you have two points, A and B,  in the plane and then determine a third point P by measuring the distances to the original two and having the sum of those distances constant, you define an ellipse.

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!

An Interactive Version of the Triangle Investigation

In the previous post, I captured movies of my investigation with The Geometer's Sketchpad. Web Sketchpad allows for including an HTML 5 version of the sketch on a webpage, like this one.

Drag the yellow dot, currently on the triangle to trace out the various positions and area of the triangle.

 

Note that I have enforced the maximum side length of 10 in a very strange way. Can you describe what I have done? Can you do it in a similar or better way?

Where do you have to place the yellow dot in order to have an area of 0?

How can you drag the yellow dot to keep the area the same?

What other patterns do you see in the behaviour? (There are lots more in the previous post)