3-Ellipses - a Question

Given n points (called foci) in a plane, an n-ellipse is the locus of all points of the plane whose sum of distances to the n foci is a constant d (https://en.wikipedia.org/wiki/N-ellipse). Let me record a simple question on 3-ellipses.

If A, B, C are the foci of a 3-ellipse and d its distance sum, do there exist three points P1, P2, P3 on the boundary of the 3-ellipse such that the following distance relations hold:

P1A = P2B = P3C
P1B = P2C = P3A
P1C = P2A = P3B


If such a triplet of points {P1,P2,P3} necessarily exists on the 3-ellipse for any non-collinear A, B, C and d, how will the triplet {P1,P2,P3} evolve when the 3-ellipse is continuously changed by changing d? What curve does each of P1, P2, P3 trace? Do such triplets have any interesting properties concerning, say, the local curvature of the 3-ellipse?
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Note: This is known to me... Pairs of points P1,P2 such that: distance(P1,A) = distance(P2,A), distance(P1,B) = distance (P2, C) and distance(P1,C) = distance(P2,B) exist and as d is tuned, both P1 and P2 trace straight lines.
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Update (April 9th, 2019). Experiments indicate that for any fixed non-collinear point set A,B,C, there is at least one 3-ellipse focused at these 3 points such that a triplet of points {P1,P2,P3} lying on the 3-ellipse satisfies the above set of distance relations. Indications are also that if triangle ABC has no symmetry, there is only one such ellipse for a given {A,B,C} and so there is no non-trivial evolution of the triplet {P1,P2,P3} when A,B,C are held fixed and only the distance sum d is changed.