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.
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.)
Which is the locus of the centers of the circles 0123 as L moves around H?
Antreas P. Hatzipolakis, 6 April 2013