HYACINTHOS 24377

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle, P, Q be two isogonal conjugate points and A'B'C' the pedal triangle of P.

 

Denote:

A", B", C" = the reflections of A', B', C' in PQ, resp.

Na, Nb, Nc = the NPC centers of A"B'C', B"C'A', C"A'B', resp.

The points Na, Nb, Nc are collinear and the line NaNbNc is perpendicular to the line PQ.

Which are the points of intersection of the OH line (Euler line) and the line NaNbNc for

1. P = H

2. P = O

?

Locus:

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

Denote:

A", B", C" =  the reflections of A', B', C' in the Euler line, resp.
Na, Nb, Nc = the NPC centers of A"B'C', B"C'A', C"A'B', resp.

Which is the locus of P such that Na, Nb, Nc are collinear?

 

[Angel Montesdeoca]:

Dear Antreas,

The locus of P such that Na, Nb, Nc are collinear is the cubic K187 of
the catalogue of Bernard Gibert (locus of foci of inscribed conics
centered on the Euler line) and a circum-quintic through the points
X(74), X(1304)

The points of intersection of the OH line (Euler line) and the line
NaNbNc for:

1. P = H is W on lines {2,3} and {143,523}

W = (2 a^14 (b^2+c^2)
-3 a^12 (3 b^4+2 b^2 c^2+3 c^4)
+5 a^10 (3 b^6+b^4 c^2+b^2 c^4+3 c^6)
-a^8 (10 b^8+3 b^6 c^2-2 b^4 c^4+3 b^2 c^6+10 c^8)
+2 a^6 (5 b^8 c^2-4 b^6 c^4-4 b^4 c^6+5 b^2 c^8)
+a^4 (b^2-c^2)^2 (3 b^8-8 b^6 c^2-8 b^2 c^6+3 c^8)
-a^2 (b^2-c^2)^4 (b^6-3 b^4 c^2-3 b^2 c^4+c^6)
-b^2 c^2 (b^2-c^2)^6+: ... : ...),

with (6-9-13)-search number (0.348196355837922,
-0.522242117838248, 3.84151070694720).

2. P = O is X(140) =midpoint of X(3) and X(5)

Best regards
Angel Montesdeoca