A SEQUENCE OF POINTS

hL Line:
Let ABC be a triangle, L a line passing through H, A'B'C' the pedal triangle of H (orthic triangle), A"B"C" the circumcevian triangle of H wrt A'B'C' (aka Euler triangle) and A1,B1,C1 the orthogonal projections of A,B,C, on L, resp. The lines A"A1, B"B1, C"C1 concur at point H11 on the pedal circle of H (NPC of ABC).
Let La,Lb,Lc be the parallels through A,B,C, to A"A1, B"B1, C"C1, resp. They are concurrent at point H12 on the circumcircle of ABC.
The line H11H12 passes through H. Let's call it hL

Concurrent Euler Lines:
Let A2,B2,C2 be the orthogonal projections of A,B,C on hL. The Euler lines of AA1A2, BB1B2, CC1C2 are concurrent at point P1.
Similarly we define hhL, hhhL...... h^nL, and we get a sequence of points Pn: P1 from L and hL, P2 from hL and hhL, etc

The parallels through A,B,C to the Euler lines of AA1A2, BB1B2, CC1C2 concur at point Q1.

Antreas P. Hatzipolakis, 16 Feb. 2013