>Let's us come back to Six Circles Theorem configuration. I
>have some things interesting and will send you when I finish
>all.
>Best regards,
>Bui Quang Tuan

Dear Alex,

When playing the applet of Six Circles Theorem configuration we can see one special point when three circles C1, C3, C5 or C2, C4, C6 are concurrent at it. Now we try to find what is this point.
We denote C(I) as a concyclic circle of six touching points. Incenter I is center of C(I).
Take a circle C1 with touching points T12 on A1A2 and T13 on A1A3.
A circle C2 shares with C1 touching point T12.
T23 is touching point of C2 with A2A3
A circle C6 shares with C1 touching point T13.
T32 is touching point of C6 with A2A3
Radical center of three circles C1, C2, C6 is A1 since A1 is intersection of two tangent lines. Hence radical axis of two circles C2, C6 is a line connected A1 and midpoint M1 of T23T32 (radical axis of two circles passes midpoint of two touching points of common tangent line). In the circle C(I), T23T32 is one chord, hence IM1 is perpendicular with A2A3, so M1 is touching point of incircle with A2A3. It means radical axis of C2, C6 (line A1M1) passes through Gergonne point Ge of A1A2A3.
Similarly at the end we can prove that radical center of C2, C4, C6 is Gergonne point, and radical center of C1, C3, C5 is also Gergonne point.
When changing the configuration, the radical center of triple of circles is fixed, hence when they concurrent, the concurrent point is Gergonne point. There are two triples but only one concurrent point Ge.

So our configuration can be used as one solution for following problem:
Given a triangle, to construct three circles through a common point, each tangent to two sides of the triangle, such that the 6 points of contact are concyclic.
This is problem of Thebault - Eves, AMM E457.
I do not have original text of the problem and read it from Paul Yiu book (page 135):
http://math.fau.edu/Yiu/EuclideanGeometryNotes.pdf

We can see also one another six concyclic points when there is one triple of circles cutting side lines of A1A2A3. In this case, six cutting points are concyclic.
When three circles intersect side lines of A1A2A3, they also intersect each other on three radical axis A1Ge, A2Ge, A3Ge. Six cutting points are concyclic by the theorem of Six Concyclic Points:
http://www.cut-the-knot.org/Curriculum/Geometry/SixConcyclicPoints.shtml

Best regards,
Bui Quang Tuan