In the applet, draggable point M defines the directions of the equilateral triangles.
We can "see" (without any proof) that :
The locus of G is a circle, going through the Fermat-Toricelli point F of ABC.
The locus of G' is a straight line
The locus of M" is an hyperbola
The locus of G" is an hyperbola.
Of course, only portions of these locii are really used, depending on the conditions
(on sides or extensions) and the circles being really inside, or outside.
For locus of M" and G", the applet "jumps" when the MNP triangle collapses into the Fermat-Toricelli point. The circle to consider is the pink circle with center M" and the "associated" blue circle.
This associated circle comes in the construction of M" from M' when choosing the other side of line AB. (see the general construction)
They swap "suddenly" when M goes through F, that is M' goes "through" infinity. The applet displays then a spurious transversal line.
The other branch of the hyperbola appears when the assiociated blue circles are all three "outside" tangent to ABC.
This property is general for any circumscribed triangle MNP of constant shape, and any noticeable point in MNP.
Draggable points A, B, C define triangle ABC.
The shape of MNP is defined through the draggable intersection point F common to the circles locus of M,N,P, hence defining the angles of this triangle.
M is draggable on its circle to rotate triangle MNP.
An arbitrary triangle mnp is used to define the barycentric coordinates of G in MNP by dragging point g.
Miquel Theorem :
Let M,N,P three points on the sides of ABC.
The circumcircles to ANP, BMP and CMN have a common point S
We then deduce the following corollary :
All similar triangles inscribed in ABC have the same Miquel point S,
which is the similitude center of all these triangles.
Here, all the equilateral triangles MNP inscribed in ABC have the same Miquel point S,
and S is the similitude center of any two such triangles.
From the MNP point of view, point S is the only point from where we see the sides of MNP under angles π-A, π-B, π-C.
Hence there is a constant similitude of center S transforming M into N. It transforms line BC into AC.
In the same way, there is a constant similitude of center S which transforms M into P.
Any point Q with constant barycentric coordinates in MNP is then obtained from M by some constant similitude of center S.
The locus of this point Q is then the transformed of locus of M, that is line BC, by that similitude, this locus is therefore a straight line.
In the applet, a such point Q is defined by the yellow points with constant ratios qM/qN and QP/Qq.
Specifically the centroid G of MNP, with coordinates (1,1,1) in MNP, describes a straight line.
This line is at easiest defined by any two positions of MNP.