HYACINTHOS 26299

[Antreas P. Hatzipolakis]:

Let ABC be a triangle and A'B'C' the pedal triangle of I.

Denote:

Na, Nb, Nc = the NPC centers of IBC, ICA, IAB, resp.

N1, N2, N3 = the reflections of Na, Nb, Nc in AI, BI, CI, resp.

1. N1N2N3 and A'B'C' share the same orthocenter and Euler line ( = OI line of ABC)

Which points of N1N2N3 are the points O, I of ABC?

2. A'B'C', N1N2N3 are: 

 

2.1 perspective.

Perspector? 

 

2.2. Circumorthologic
(ie the orthologic center (A'B'C', N1N2N3)  lies on the circumcircle of A'B'C' ( = incircle of ABC)
and the orthologic center (N1N2N3, A'B'C')  lies on the circumcircle of N1N2N3)

 

2.3. circumparallelogic

(ie the parallelogic center (A'B'C', N1N2N3)  lies on the circumcircle of A'B'C' ( = incircle of ABC)
and the parallelogic center (N1N2N3, A'B'C')  lies on the circumcircle of N1N2N3)

Parallelogic center (A'B'C', N1N2N3) = antipode in the circumcircle of A'B'C' of the orthologic center (A'B'C', N1N2N3)

Parallelogic center ( N1N2N3, A'B'C') = antipode in the circumcircle of N1N2N3 of the orthologic center (N1N2N3, A'B'C')

[Peter Moses]:

 

 

Hi Antreas,

1) X(23) & X(186).

2.1)

a (a+b-c) (a-b+c) (a^3-a b^2-2 a b c+b^2 c-a c^2+b c^2) (a^2 b^2-b^4-2 a^2 b c+a b^2 c+a^2 c^2+a b c^2+2 b^2 c^2-c^4)::  

on lines {52,517}.

Searches: {4.13338338572036404674876394915,1.12424314033219476230738301594,3.65495292932745110307185294632}.

2.2)

(A' B'C', N1N2N3) orthology: 

 

a^2 (a+b-c) (a-b+c) (a^2 b-b^3+a^2 c-4 a b c+2 b^2 c+2 b c^2-c^3)^2:: 

on lines {{1,3025},{11,517},{12,3259},{ 55,953},{56,901},{513,1317},{ 1155,5577},{1319,1357},{1364, 5048},{3028,4017},{3057,3326}, {3328,5919},{3878,7144}}.

reflection of X(3025) in X(1).

On the incircle.

reflection of X(1317) in the OI line.

Searches: {0. 042013400200665666126165386337 5,1. 22493634414447922692196915598, 2. 77324082817635548472953166236} .

 

(N1N2N3, A'B'C') orthology: X(3244).

2.3)

(A'B'C',  N1N2N3) parallelogy: X(3025).

(N1N2N3,  A'B'C') parallelogy: X(946).

Best regards,

Peter Moses.