CIRCUMCENTER OF N1N2N3

Let ABC be a triangle, P a point and N1,N2,N3 the NPC centers of PBC, PCA, PAB, resp.

Which is the circumcenter Op of N1N2N3 ?

P = H. Op = N [N1 = N2 = N3 = N (N1N2N3 is degenerated)]

P = I. Op = N

P = O. Op lies on the line NU, where U is the Poncelet point of O wrt ABC (ie the point where concur the NPCs of OBC,OCA,OAB and ABC)

P = N. Op lies on the Euler line of ABC.

If P describes a line (Euler line, etc) which is the locus of Op?

If P describes the circumcircle, Op is the Poncelet point of P wrt ABC. If P describes other curves?

Antreas P. Hatzipolakis, 14 April 2013

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For P = O, Op = complement of X(157).

In general, Op = complement of the nine-point-center [N'] of the antipedal triangle [A'B'C'] of P.

Randy Hutson, Hyacinthos #21963