- Dear Antreas and Paul,

[APH]: Let ABC be a triangle P a point and PaPbPc the pedal

triangle of P.

Let Ha, Hb, Hc be the orthocenters of the triangles

APbPc, BPcPa, CPaPb, resp.

Which is the locus of P such that PaPbPc, HaHbHc

are perspective?

If the locus is the whole plane, then what is the locus

of the perspectors as P moves on the Euler line?

[PY]: It is indeed the whole plane. The two triangles are

oppositely congruent. If P = (u:v:w) in homogeneous barycentric

coordinates, then the homothetic center is the point

(a^2(b^2c^2u + c^2S_C v + b^2S_B w) : ... : ...).

For P = O, this is the nine-point center, and

for P = H, this is the center of the Taylor circle X(389).

***********

A little synthetic.

If Ma, Mb, Mc are the mid points of the sides of the pedal triangle

and Pg is the centroid of the pedal triangle then Ma is the mid point

of PHa and the mid point Q of PaHa is the complement of P in the

pedal triangle or Q is produced from P in the homothety (Pg, -1/2).

So PaHa, PbHb, PcHc are condurrent at Q and the triangles

pedal of P and HaHbHc are oppositely congruent.

For P = O then Pg = G and Q = X(5)

for P = H then Pg = X(51) and Q = X(389)

Best regards

Nikolaos Dergiades - Let ABC be a triangle and A'B'C', A"B"C" the pedal triangles

of two isogonal conjugate points P,P*.

Denote:

Ha := The orthocenter of A'B"C"

Hb := The orthocenter of B'C"A"

Hc := The orthocenter of C'A"B"

Are the triangles ABC, HaHbHc perspective + parallelogic + orthologic

for all P's? (If no, then for which P's are they?)

Perspector, and parallelo/orthologic centers ?

Antreas - Dear Antreas
> Let ABC be a triangle and A'B'C', A"B"C" the pedal triangles

ABC and HaHbHc are indirectly similar; hence, they are parallelogic and orthologic (two of the centers are upon the circumcircle of ABC, the two other ones upon the circumcircle of HaHbHc)

> of two isogonal conjugate points P,P*.

>

> Denote:

>

> Ha := The orthocenter of A'B"C"

>

> Hb := The orthocenter of B'C"A"

>

> Hc := The orthocenter of C'A"B"

>

> Are the triangles ABC, HaHbHc perspective + parallelogic + orthologic

> for all P's? (If no, then for which P's are they?)

>

> Perspector, and parallelo/orthologic centers ?

I think that they are perspective only when P lies on the Neuberg cubic.

Friendly. Jean-Pierre - Let ABC be a triangle, P = (x:y:z) a point and A'B'C' the

cevian triangle of P.

Denote:

Ab := BC /\ (perpendicular from A to BB')

Ac:= BC /\ (perpendicular from A to CC')

Similarly Bc, Ba and Ca, Cb.

Let A*, B*, C* be the orthocenters of AAbAc, BBcBa, CCaCb resp.

ABC, A*B*C* are orthologic by construction.

Which is the other than H orthologic center?

If P = I, then the orth. center lies on the circumcircle of ABC.

Which is the locus of P such that the other orthologic

center is on the circumcircle of ABC?

Antreas