HYACINTHOS 26716

 [Antreas P. Hatzipolakis]:

 

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

Denote:

Na, Nb, Nc = the NPC centers of PBC, PCA, PAB, resp.

A", B", C" = the orthogonal projections of Na, Nb, Nc on AA', BB', CC', resp.

N1, N2, N3 = the NPC centers of A"BC, B"CA, C"AB, resp.

Which is the locus of P such that ABC, N1N2N3 are orthologic ?

 

The entire plane? If yes:

1. Which are the orthologic centers in terms of P?

2. Which are the loci of the orthologic centers as P moves on the Euler line?

 

 

[Peter Moses]:

Hi Antreas,

1)
(ABC, N1N2N3) = a^2 p ((a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) p^2+a^2 (a^2-b^2+c^2) p q+a^2 (a^2+b^2-c^2) p r+2 a^4 q r) ((2 a^2+b^2-2 c^2) (a^2-b^2-c^2) p q+(a^4+a^2 b^2-2 b^4-2 a^2 c^2+b^2 c^2+c^4) q^2+2 b^4 p r+(2 a^2-b^2-2 c^2) (a^2+b^2-c^2) q r) ((2 a^4-5 a^2 b^2+3 b^4-4 a^2 c^2-3 b^2 c^2+2 c^4) p q+(a^4-3 a^2 b^2+2 b^4-2 a^2 c^2-3 b^2 c^2+c^4) q^2+2 b^4 p r+(2 a^4-3 a^2 b^2+3 b^4-4 a^2 c^2-5 b^2 c^2+2 c^4) q r) (2 c^4 p q+(a^2-b^2-c^2) (2 a^2-2 b^2+c^2) p r+(2 a^2-2 b^2-c^2) (a^2-b^2+c^2) q r+(a^4-2 a^2 b^2+b^4+a^2 c^2+b^2 c^2-2 c^4) r^2) (2 c^4 p q+(2 a^4-4 a^2 b^2+2 b^4-5 a^2 c^2-3 b^2 c^2+3 c^4) p r+(2 a^4-4 a^2 b^2+2 b^4-3 a^2 c^2-5 b^2 c^2+3 c^4) q r+(a^4-2 a^2 b^2+b^4-3 a^2 c^2-3 b^2 c^2+2 c^4) r^2)::

(N1N2N3, ABC) = X(5).

For P = O

(ABC, N1N2N3) = a^2 (a^2-b^2-c^2) (a^6-a^4 b^2-a^2 b^4+b^6-3 a^4 c^2-a^2 b^2 c^2-3 b^4 c^2+3 a^2 c^4+3 b^2 c^4-c^6) (a^6-3 a^4b^2+3 a^2 b^4-b^6-a^4 c^2-a^2 b^2 c^2+3 b^4 c^2-a^2 c^4-3 b^2 c^4+c^6):: 

on lines {{2,13418},{4,13585},{6,3205}, {54,5946},{68,10255},{6

 

Best regards,

Peter Moses.