Σάββατο, 6 Απριλίου 2013

ORTHOPOLAR CIRCLES [orthic triangle]

Let ABC be a triangle and A'B'C' the orthic triangle.

Denote:

Ab,Ac = the reflections of A' in BB',CC', resp.

Bc,Ba = the reflections of B' in CC',AA', resp.

Ca,Cb = the reflections of C' in AA',BB', resp.

Let L be a line passing through H.

Denote:

0 = the orthopole of L wrt A'B'C'

1 = the orthopole of L wrt A'AbAc

2 = the orthopole of L wrt B'BcBa

3 = the orthopole of L wrt C'CaCb

The points 0,1,2,3 are concyclic.

Special Case: L = Euler line of ABC.

L passes through the common circumcenter of the triangles A'AbAc, B'BcBa, C'CaCb [= the H of ABC] and the circumcenter of A'B'C' [=the N of ABC]. The points 0,1,2,3 coincide with the Poncelet point U of H wrt A'B'C'. (The U is the point of concurrence of 7 NPCs: The NPCs of A'B'C', HB'C', HC'A', HA'B', A'AbAc, B'BcBa, C'CaCb.)

Problem:

Which is the locus of the centers of the circles 0123 as L moves around H?

Antreas P. Hatzipolakis, 6 April 2013

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