HYACINTHOS 28914

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle.

Denote:

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

A', B', C' = the orthogonal projections of Na, Nb, Nc on IA, IB, IC, resp.

La, Lb, Lc = the reflections of the OI line in A'Na, B'Nb, C'Nc, resp.

A*B*C* = the triangle bounded by La, Lb, Lc

1. ABC, A*B*C* are orthologic

2. ABC, A*B*C* are parallelogic

[Peter Moses]:

 

Hi Antreas,

1) Orthologic.

(ABC, A*B*C*) =  X(100).

 

(A*B*C*, ABC) = X(3)X(10)∩X(4)X(1647)

  

= a^6 b+3 a^5 b^2-a^4 b^3-4 a^3 b^4+a^2 b^5+a b^6-b^7+a^6 c-10 a^5 b c+2 a^4 b^2 c+8 a^3 b^3 c-3 a^2 b^4 c+2 a b^5 c+3 a^5 c^2+2 a^4 b c^2-8 a^3 b^2 c^2+2 a^2 b^3 c^2-a b^4 c^2+2 b^5 c^2-a^4 c^3+8 a^3 b c^3+2 a^2 b^2 c^3-4 a b^3 c^3-b^4 c^3-4 a^3 c^4-3 a^2 b c^4-a b^2 c^4-b^3 c^4+a^2 c^5+2 a b c^5+2 b^2 c^5+a c^6-c^7 : : 

 

= lies on these lines: {3,10}, {4,1647}, {5,25377}, {946,3667}, {2827,25437}, {3976,30384}

2) Parallelogic.

(ABC,  A*B*C*) = X(104).

 

(A*B*C*,  ABC) = X(1)X(2)∩X(522)X(946)

  

= a^6 b+a^5 b^2-3 a^4 b^3-2 a^3 b^4+3 a^2 b^5+a b^6-b^7+a^6 c-6 a^5 b c+4 a^4 b^2 c+6 a^3 b^3 c-5 a^2 b^4 c+a^5 c^2+4 a^4 b c^2-8 a^3 b^2 c^2+2 a^2 b^3 c^2-a b^4 c^2+2 b^5 c^2-3 a^4 c^3+6 a^3 b c^3+2 a^2 b^2 c^3-b^4 c^3-2 a^3 c^4-5 a^2 b c^4-a b^2 c^4-b^3 c^4+3 a^2 c^5+2 b^2 c^5+a c^6-c^7 : : 

 

= lies on these lines: {1,2}, {522,946}, {3738,25437}

Best regards,
Peter Moses.