Dear Mariano Perez de la Cruz,

Thank you for your kind words and interesting with our problem! It is very interesting if we can create some more mechanisms from this geometric essence.

After careful studying your idea, I am not sure that I understand you well.

We are talking about two configurations:

Eves' configuration:
http://www.cut-the-knot.org/Curriculum/Geometry/ExpandedIncircle.shtml

and my configuration:
http://www.cut-the-knot.org/Curriculum/Geometry/BuiQuangTuan.shtml

Both configurations generate six concyclic points on sidelines of a triangle. Eves created these six points as common intersection points of two circles centered at two vertices of the triangle. I created these six points as common tangent points of two circles each touching two sidelines of the triangle.

If you use symbol names (points, lines, circle...) in one of two above configurations then it is better to understand. May be you can also create your image and send us please!

In any case, I am worry about two matters:
1. Twelve travellers (four at each vertex) are may be too much?
2. Where their derpart positions? As you can play in two configurations (of Eves and of mine), derpart positions are at three touching points of incircle, so not at the vertices as you design.

I am very sorry if I understand you not very well!
Best regards,
Bui Quang Tuan
>
>Let us asume that we have twelve travellers as follow
>4 located on vertex A we shall call them A1b , A2c, A2b and
>A3c
>4 located on vertex B we shall call them B1c, B1c, B2a and
>B2c
>4 located on vertex C we shall call them C1b, C1a, C2b and
>C2a
>
>The first subindex take into account the generation they
>belong and the second subindex for the destination vertex
>
>
>At the instant T=0 all the travellers start to depart to the
>each destination vertex,