HYACINTHOS 26298

[Antreas P. Hatzipolakis]:

Let ABC be a triangle.

Denote:

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

Naa, Nab, Nac = the reflections of Na in AI, BI, CI, resp.

Nba, Nbb, Nbc = the reflections of Nb in AI, BI, CI, resp.

Nca, Ncb, Ncc = the reflections of Nc in AI, BI, CI, resp.

The NPCs of NaaNabNac, NbaNbbNbc,  NcaNcbNcc are concurrent at a point on the OI line.

Denote:

A'B'C' = the pedal triangle of I

N1, N2, N3 = the NPC centers of NaaNabNac, NbaNbbNbc,  NcaNcbNcc, resp.

The triangles A'B'C', N1N2N3 are homothetic (and share the same Euler line = OI line of ABC. I is the orthocenter of N1N2N3)

The homothetic center lies on the OI line of ABC.

The triangles NaaNbbNcc, N1N2N3 are: 

1. circumorthologic

(ie the orthologic center (NaaNbbNcc, N1N2N3)  lies on the circumcircle of NaaNbbNcc
and the orthologic center (N1N2N3, NaaNbbNcc)  lies on the circumcircle of N1N2N3)

 

2. circumparallelogic

(ie the parallelogic center (NaaNbbNcc, N1N2N3)  lies on the circumcircle of NaaNbbNcc
and the parallelogic center (N1N2N3, NaaNbbNcc)  lies on the circumcircle of N1N2N3)

Parallelogic center (NaaNbbNcc, N1N2N3) = antipode in the circumcircle of NaaNbbNcc of the orthologic center (NaaNbbNcc, N1N2N3)

Parallelogic center ( N1N2N3, NaaNbbNcc) = antipode in the circumcircle of N1N2N3 of the orthologic center (N1N2N3, NaaNbbNcc)

 

[César Lozada]:

 

> The NPCs of NaaNabNac, NbaNbbNbc,  NcaNcbNcc are concurrent at a point on the OI line.

 

At X(942)

 

> The triangles A'B'C', N1N2N3 are homothetic

Z = ((b+c)*a*(a^2-b^2-b*c-c^2)+(b- c)^2*((b+c)^2-a^2))*(a-b+c)*(a +b-c) : :  (trilinears)

= (4*R^2-3*r^2-5*s^2+5*SW)*X(1)+ (9*r^2-s^2+SW)*X(3)

= On lines:  {1,3}, {7,10266}, {11,12005}, {12,5083}, {244,2594}, {1317,3754}, {1421,8614}, {2801,7173}, {3678,7294}, {3874,5433}, {5253,12739}, {5883,10944}, {5901,11570}

= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (942,1319,65), (942,5045,5425), (5570,13373,2646)

= [ 3.408689900362678, 3.10922094427124, -0.085114587370791

 

>1. circumorthologic

O(Naa->N1) = X(3244)

 

O(N1->Naa) = (b+c)*a^5-2*(b^2+c^2)*a^4-(b+ c)*(3*b^2-8*b*c+3*c^2)*a^3+(3* b^4+3*c^4-2*b*c*(b^2+3*b*c+c^ 2))*a^2+(b+c)*(2*b^4+2*c^4-b* c*(8*b^2-13*b*c+8*c^2))*a-(b^ 2-c^2)^2*(b-c)^2 : :  (trilinears)

= On lines: {1,901}, {513,3754}, {517,548}

= [ 1.399204864330000, 1.98453857357000, 1.620966301280000 ]

 

>2. circumparallelogic

P(Naa->N1) = X(946)

 

P(N1->Naa) = (b+c)*a^8-(b+c)^2*a^7-2*(b^3+ c^3)*a^6+3*(b^2+c^2)^2*a^5-3* b^2*c^2*(b+c)*a^4-(3*b^4+5*b^ 2*c^2+3*c^4)*(b-c)^2*a^3+(b^2- c^2)*(b-c)*(2*b^4+2*c^4-b*c*( 2*b^2-3*b*c+2*c^2))*a^2+(b^2- c^2)*(b-c)*(b^3+c^3)*(b^2-3*b* c+c^2)*a-(b^2-c^2)^3*(b-c)^3

= On lines: {1,953}, {513,12005}, {517,1125}

= [ 3.366857151000000, 2.54007659591000, 0.328215845810000 ]

 

César Lozada