HYACINTHOS 29569

[Antreas P. Hatzipolakis]:

Let ABC be a triangle.

Denote:

Na, Nb, Nc = the NPC centers of IBC, ICA, IAB, resp.
N1, N2, N3 = the reflections of Na, Nb, Nc in BC, CA, AB, resp.

Ab, Ac = the orthogonal projections of N2, N3 on AI, resp.
Bc, Ba = the orthogonal projections of N3, N1 on BI, resp.
Ca, Cb = the orthogonal projections of N1, N2 on CI, resp.

We have:
Ab = Ac =: A'
Bc = Ba =: B'
Ca = Cb =: C'

1. A'B'C', ABC are orthologic
2. A'B'C', NaNbNc are orthologic.
Orthologic centers? 

 
Also:
3. A'B'C', N1N2N3 are orthologic by construction (A'B'C' is the pedal triangle of I wrt triangle N1N2N3)
Orthologic center (A'B'C', N1N2N3) = I
The orhologic center (N1N2N3, A'B'C') is the isogonal conjugate of I wrt N1N2N3
Point wrt triangle ABC?


[Peter Moses]:


HI Antreas,

1.
(ABC, A'B'C') = X(1389).
(A'B'C', ABC) = X(31870).

2.
(A'B'C',  NaNbNc) = X(6797).

(NaNbNc, A'B'C')  =

 X(1)X(140) ∩ X(5)X(3884) =  

= 2*a^6*b - 3*a^5*b^2 - 3*a^4*b^3 + 6*a^3*b^4 - 3*a*b^6 + b^7 + 2*a^6*c - 12*a^5*b*c + 11*a^4*b^2*c + 7*a^3*b^3*c - 12*a^2*b^4*c + 5*a*b^5*c - b^6*c - 3*a^5*c^2 + 11*a^4*b*c^2 - 20*a^3*b^2*c^2 + 12*a^2*b^3*c^2 + 3*a*b^4*c^2 - 3*b^5*c^2 - 3*a^4*c^3 + 7*a^3*b*c^3 + 12*a^2*b^2*c^3 - 10*a*b^3*c^3 + 3*b^4*c^3 + 6*a^3*c^4 - 12*a^2*b*c^4 + 3*a*b^2*c^4 + 3*b^3*c^4 + 5*a*b*c^5 - 3*b^2*c^5 - 3*a*c^6 - b*c^6 + c^7 : :
= 2 X[140] + X[5559], X[1483] + 2 X[15862]

= lies on these lines: {1, 140}, {5, 3884}, {10, 1484}, {21, 952}, {495, 13375}, {515, 26202}, {517, 5499}, {519, 31650}, {632, 8256}, {1145, 34126}, {1385, 32905}, {1389, 5901}, {1482, 6853}, {1483, 15862}, {4187, 7705}, {5450, 32613}, {6583, 11362}, {6842, 10129}, {6940, 22765}, {7508, 10944}, {11231, 33895}, {16159, 28212}, {19524, 32141}, {19907, 31659}

= reflection of X(1389) in X(5901)

3.
(N1N2N3, A'B'C')  =

 X(30)X(17643) ∩ X(65)X(5844) =  

= a*(a^8*b - 2*a^7*b^2 - 2*a^6*b^3 + 6*a^5*b^4 - 6*a^3*b^6 + 2*a^2*b^7 + 2*a*b^8 - b^9 + a^8*c - 6*a^7*b*c + 9*a^6*b^2*c + 2*a^5*b^3*c - 17*a^4*b^4*c + 14*a^3*b^5*c + 3*a^2*b^6*c - 10*a*b^7*c + 4*b^8*c - 2*a^7*c^2 + 9*a^6*b*c^2 - 24*a^5*b^2*c^2 + 19*a^4*b^3*c^2 + 15*a^3*b^4*c^2 - 27*a^2*b^5*c^2 + 11*a*b^6*c^2 - b^7*c^2 - 2*a^6*c^3 + 2*a^5*b*c^3 + 19*a^4*b^2*c^3 - 40*a^3*b^3*c^3 + 22*a^2*b^4*c^3 + 10*a*b^5*c^3 - 11*b^6*c^3 + 6*a^5*c^4 - 17*a^4*b*c^4 + 15*a^3*b^2*c^4 + 22*a^2*b^3*c^4 - 26*a*b^4*c^4 + 9*b^5*c^4 + 14*a^3*b*c^5 - 27*a^2*b^2*c^5 + 10*a*b^3*c^5 + 9*b^4*c^5 - 6*a^3*c^6 + 3*a^2*b*c^6 + 11*a*b^2*c^6 - 11*b^3*c^6 + 2*a^2*c^7 - 10*a*b*c^7 - b^2*c^7 + 2*a*c^8 + 4*b*c^8 - c^9) : :

= lies on these lines: {30, 17643}, {65, 5844}, {517, 5499}, {1385, 3754}, {1389, 5253}, {1483, 5885}, {5690, 15844}, {5883, 33657}

Best regards,
Peter Moses.