the study of a curve to 3 points with same sum of distances:case 1

We all know that a curve to two points with same sum of distances is ellipse.

But how about to 3 points?

I try to study this case as I haven't find out any mention about it.

My study show that the curve is interesting, I start from ths most simple case:

3points in one line and with equal distance c , the sum of the distance is p and one important figure is, p=3c,

another important figure is p=2c which is limitation of the p. if p=2c, that means the curve is a point.

figure1 is p<3c,>> 2c,
figure 2 is p=2c,which like an eye
figure 3 is p>3c, similar as fig1
figure 4 is p very close to 2c, which like a ellipse

the blue curve is real curve and brown curve is ellipse which simply for reference.