22475Another Inversion Th. ? [Re: Generalization of P. Moses Circle (Re: Trilinear poles)]

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• Jun 28, 2014
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Another Inversion Theorem (?)

Let (A), (B), (C) be three circles tangent all externally or internally
a circle Q at A',B',C', resp.

Denote:

Ab = the inversion of A' in (B)
Ac = the inversion of A' in (C)
Bc = the inversion of B' in (C)
Ba = the inversion of B' in (A)
Ca = the inversion of C' in (A)
Cb = the inversion of C' in (B)

Ma = the midpoint of BaCa
Mb = the midpoint of CbAb
Mc = the midpoint of AcBc

The lines AMa, BMb, CMc are concurrent ??

If true, then we can apply it to Circumcircle and three Lucas Circles,
to NPC and three excircles etc

Antreas

[APH]

Let (A), (B), (C) be three externally tangent circles.
The [Apollonian] circle tangent to (A),(B),(C) internally
touches them at A',B',C', resp.

Denote:

Ab = the inversion of A' in (B)
Ac = the inversion of A' in (C)
Bc = the inversion of B' in (C)
Ba = the inversion of B' in (A)
Ca = the inversion of C' in (A)
Cb = the inversion of C' in (B)

Are the six points Ab,Ac.Bc,Ba,Bc,Ca,Cb always concyclic?

In my figure they are, and the center of the circle lies
on the line joining the circumcenters of ABC and A'B'C'.

Is it true? And if yes, is it already known? (references?)

Antreas