Thear Antreas,

> [APH]

> > Let ABC be a triangle, HaHbHc the pedal triangle of H

> > (orthic triangle), P a point, A1B1C1 the circumcevian triangle

> > of P and A2,B2,C2 the reflections of A1,B1,C1 in Ha,Hb,Hc,

> > resp.

> > Which is the locus of P such that the circumcenter of

> > A2B2C2 is lying on the Euler Line of ABC?

> >

> > It is the Euler Line + ?????

This is the Euler line + A quintic though A, B, C, H, Ha, Hb, Hc

Equation:

-a^4 c^2 x^3 y^2 + 2 a^2 b^2 c^2 x^3 y^2 - b^4 c^2 x^3 y^2 +

a^2 c^4 x^3 y^2 - b^2 c^4 x^3 y^2 - a^4 c^2 x^2 y^3 +

2 a^2 b^2 c^2 x^2 y^3 - b^4 c^2 x^2 y^3 - a^2 c^4 x^2 y^3 +

b^2 c^4 x^2 y^3 + a^6 x^3 y z - 2 a^4 b^2 x^3 y z +

a^2 b^4 x^3 y z - 2 a^4 c^2 x^3 y z + 4 a^2 b^2 c^2 x^3 y z -

2 b^4 c^2 x^3 y z + a^2 c^4 x^3 y z - 2 b^2 c^4 x^3 y z +

a^6 x^2 y^2 z - a^4 b^2 x^2 y^2 z - a^2 b^4 x^2 y^2 z +

b^6 x^2 y^2 z - 2 a^4 c^2 x^2 y^2 z + 4 a^2 b^2 c^2 x^2 y^2 z -

2 b^4 c^2 x^2 y^2 z + c^6 x^2 y^2 z + a^4 b^2 x y^3 z -

2 a^2 b^4 x y^3 z + b^6 x y^3 z - 2 a^4 c^2 x y^3 z +

4 a^2 b^2 c^2 x y^3 z - 2 b^4 c^2 x y^3 z - 2 a^2 c^4 x y^3 z +

b^2 c^4 x y^3 z - a^4 b^2 x^3 z^2 + a^2 b^4 x^3 z^2 +

2 a^2 b^2 c^2 x^3 z^2 - b^4 c^2 x^3 z^2 - b^2 c^4 x^3 z^2 +

a^6 x^2 y z^2 - 2 a^4 b^2 x^2 y z^2 + b^6 x^2 y z^2 -

a^4 c^2 x^2 y z^2 + 4 a^2 b^2 c^2 x^2 y z^2 - a^2 c^4 x^2 y z^2 -

2 b^2 c^4 x^2 y z^2 + c^6 x^2 y z^2 + a^6 x y^2 z^2 -

2 a^2 b^4 x y^2 z^2 + b^6 x y^2 z^2 + 4 a^2 b^2 c^2 x y^2 z^2 -

b^4 c^2 x y^2 z^2 - 2 a^2 c^4 x y^2 z^2 - b^2 c^4 x y^2 z^2 +

c^6 x y^2 z^2 + a^4 b^2 y^3 z^2 - a^2 b^4 y^3 z^2 -

a^4 c^2 y^3 z^2 + 2 a^2 b^2 c^2 y^3 z^2 - a^2 c^4 y^3 z^2 -

a^4 b^2 x^2 z^3 - a^2 b^4 x^2 z^3 + 2 a^2 b^2 c^2 x^2 z^3 +

b^4 c^2 x^2 z^3 - b^2 c^4 x^2 z^3 - 2 a^4 b^2 x y z^3 -

2 a^2 b^4 x y z^3 + a^4 c^2 x y z^3 + 4 a^2 b^2 c^2 x y z^3 +

b^4 c^2 x y z^3 - 2 a^2 c^4 x y z^3 - 2 b^2 c^4 x y z^3 +

c^6 x y z^3 - a^4 b^2 y^2 z^3 - a^2 b^4 y^2 z^3 + a^4 c^2 y^2 z^3 +

2 a^2 b^2 c^2 y^2 z^3 - a^2 c^4 y^2 z^3

>

> Variation (easier?):

>

> Let ABC be a triangle, HaHbHc the pedal triangle of H

> (orthic triangle), P a point, and A',B', and C' the reflections

> of P in Ha, Hb, Hc, resp.

> Which is the locus of P such that the circumcenter of

> A'B'C' is lying on the Euler Line of ABC?

>

> Is it Euler Line +???

>

This is Euler line + nothing

> APH

>