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!

Finding Points of Intersection with The Geometer's Sketchpad

So Frank asked me about finding intersections of plotted linear functions at our Web 2.0 session last week. It turns out that constructing the intersection of two lines is trivial but of two function plots is not.

I told him the story about how happy I was to have written a custom tool to take two linear functions as input and create their point of intersection for Stephanie. The .gsp file is here.

At least, I was happy until I asked my pal Shawn Godin about it and he very quickly created a sketch that takes two polynomial functions (and maybe other kinds), and the origin and plots the functions, creates a locus of all the points of intersection found using Newton's Method, and labels one of them. You can then drag the point on the locus through all the points of intersection. His .gsp file is here.

Guess which one of us is working on our PhD in Mathematics?

Thanks Shawn for giving me permission to share the sketch. I'll be curious to see if this post gets as much traffic as the post about graphing inequations in Sketchpad, which also reported Shawn's guru status in the Sketchpad community.