Dear Juan Carlos

> If we consider the three small extangent circles (O_1),(O_2),(O_3)

and

> NPC of ABC, then the radicals axis to the three circles (O_1),(O_2),

> (O_3) with NPC performed a triangle QST, hence QST and ABC are

> perspective at P'.

> Is P'in ETC?

If I well understand, (O_1) is a circle touching BC and touching

externally the B-excircle and the C-excircle.

In this case, it is not necessary to choose the small one.

Let X, X' be the contact points with BC of the circles Cx and Cx'

touching BC and touching externally the B and C excircles.

Then the circle with diameter XX' is centered at Za = BC inter IbIc and

is orthogonal to the NPC.

It follows that Za is the radical center of (NPC,Cx,Cx')

As Za, Zb, Zc are on the same line, it follows from Desargues theorem

that the 8 triangles with side lines La,Lb,Lc are perspective with ABC

where La = radical axis(NPC, Cx) or radical axis(NPC, Cx'),....

Of course, it could be interesting to look at the configuration of the

8 perspectors.

Friendly. Jean-Pierre