Lemma:

Let x, y be two perpendicular lines and L a line.

The reflections of L in x, y are parallels.

Corollary:

Let ABC be a triangle, x,y,z three lines perpendicular to BC, CA, AB, resp. and L1, L2, L3 three lines.

Denote:

La, Lx = the reflections of L1 in BC, x, resp.

Lb, Ly = the reflections of L2 in CA, y, resp.

Lc, Lz = the reflections of L3 in AB, z, resp.

AaBbCc = the triangle bounded by La, Lb, Lc, resp.

AxByCz = the triangle bounded by Lx, Ly, Lz, resp.

AaBbCc, AxByCz are homothetic.

Application:

Let ABC be a triangle, P a point and A'B'C' the pedal triangle of P.

Denote:

L1, L2, L3 = the Euler lines of AB'C', BC'A', CA'B', resp.

La, Lb, Lc = the reflections of L1, L2, L3 in BC, CA, AB, resp.

Li, Lii, Liii = the reflections of L1, L2, L3 in PA', PB', PC', resp.

AaBbCc = the triangle bounded by La,Lb,Lc

AiBiiCiii = the triangle bounded by Li, Lii, Liii

Which is the locus of the homothetic center of AaBbCc, AiBiiCiii as P moves on a line, the Euler line for example?