I must congratulate you both Bui Quam and Alex for the rich and interesting research on the subject

"RE: Cyclic Touching Circles Around Triangle" posed by our friend Bui

I outline below an "engineering solution" to the original challenge , of course not rigorous as the one exhibited in CTK
but I belive is valid so I kindly ask your expertise in checking if any flaws on it.

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,
The speed of each traveller will be such that, after elapsed a given unit time U,
all Travellers simultaneously will meet by pairs as follows

A1b meet with B1a ;
B1c meet with C1b ;
C1a meet with A2c ;
A2b meet with B2a ;
B2c meet with C2b ;
C2a meet with A3c

We name the speed of the travellers as follows
For A1b equals Va1
For B1a and B1c equal Vb1
For C1b and C1a equal Vc1
For A2c and A2b equal Va2
For B2a and B2c equal Vb2
For C2b and C2a equal Vc2

After a time U he distance walked by each of above the pairs of travellers will amount respectively one side of the triangle ( La, Lb, Lc)
So we can write the following equations
(Va1+Vb1) U = Lc ; (1)
(Vb1+Vc1) U = La ; (2)
(Vc1+Va2) U = Lb ; (3)
(Va2+Vb2) U = Lc ; (4)
(Va2+Vc2) U = La ; (5)

Substracting Eq (4) from (1) and Eq (5) from (2)

We get U( Va1-Va2) = U( Vb2 -Vb1) = U( Vc1-Vc2) what means the difference between the radius of circunferences drawn from vertices A, B and C
are equals and will keep equals forever.

Now to the second part, once is proved that distance of the two radius drawn from the same vertex is constant, is obvious that midpoint of this two radius is the tangency point of the incircle since two consecutive midpoints are equidistant from the common vertex of its side.
Being the tangency point the mentioned midpoint if from the incenter we draw a circunference passing intersection of side Lc with Ra1 (UVa1) it will pass also over the other intersection with Ra2 (UVa2)