HYACINTHOS 25958

[Antreas P. Hatzipolakis]:

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

Denote:

Ab, Ac = the orthogonal projections of A' on AC, AB, resp.
A2, A3 = the reflections of B', C' in Ab, Ac, resp.

Bc, Ba = the orthogonal projections of B' on BA, BC, resp.
B3, B1 = the reflections of C', A' in Bc, Ba, resp.

Ca, Cb = the orthogonal projections of C' on CB, CA, resp.
C1, C2 = the reflections of A', B' in Ca, Cb, resp.

Na, Nb, Nc = the NPC centers of AA2A3, BB3B1, CC1C2, resp.

P = H:
ABC, NaNbNc are homothetic.
Homothetic center?
Orthologic center (NaNbNc, ABC) = orthocenter of NaNbNc ?


Which is the locus of P such that:

1. ABC, NaNbNc are perspective?
2. ABC, NaNbNc are orthologic? The Euler line?
And which are the loci of the orthologic centers as P moves on the Euler
line?


[Peter Moses]:

Hi Antreas,

>P = H:
>ABC, NaNbNc are homothetic.
>Homothetic center?

 

X(54).

>Orthologic center (NaNbNc, ABC) = orthocenter of NaNbNc ?

2 a^10-4 a^8 b^2+a^6 b^4+a^4 b^6+a^2 b^8-b^10-4 a^8 c^2+8 a^6 b^2 c^2-a^4 b^4 c^2-6 a^2 b^6 c^2+3 b^8 c^2+a^6 c^4-a^4 b^2 c^4+10 a^2 b^4 c^4-2 b^6c^4+a^4 c^6-6 a^2 b^2 c^6-2 b^4 c^6+a^2 c^8+3 b^2 c^8-c^10:: 

on lines {{3,2929},{4,54},{5,1511},{6,3 82},{20,11438},{30,143},{49,11 3},{51,6240},{115,1970},{125, 3520},...}.
Midpoint of X(i) and X(j) for these {i,j}: {{382,11750},{1885,6146}}.
Reflection of X(i) in X(j) for these {i,j}: {{389,12241},{3575,10110},{101 12,12370}}.
{X(5),X(12038)}-harmonic conjugate of X(5972).
crosssum of X(3) and X(6102).

1) circular cubic through X{4,15,16,399,2070,6243} & reflection (A in BC...) triangle.

2) Euler line + circle center X(11250) through diameter X(1147,3357).

>And which are the loci of the orthologic centers as P moves on the Euler line?

ABC NaNbNc: Jerabek.
NaNbNc ABC: line through {30,143,389,5446,6663,11692,12 241,12897}.

Best regards,
Peter Moses.