HYACINTHOS 26332

[Antreas P. Hatzipolakis]:

Let ABC be a triangle.

Denote:

(Ia), (Ib), (Ic) = the A-, B-,C- excircle, resp.

(Na), (Nb), (Nc) = the NPCs of IBC, ICA, IAB, resp.

Ra = the radical axis of (Ia) and (Na)
Rb = the radical axis of (Ib) and (Nb)
Rc = the radical axis of (Ic) and (Nc)

A*B*C* = the triangle bounded by Ra, Rb, Rc

1. The NPC of A*B*C* passes through the Feuerbach point.

Center of the circle?

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

Orthologic centers?



[Angel Montesdeoca]:

*** 1. Center of the NPC of A*B*C* :

W = ( a^8 (b-c)^2
         -2 a^7 (b^3+5 b^2 c+5 b c^2+c^3)
          -a^6 (2 b^4+11 b^3 c+30 b^2 c^2+11 b c^3+2 c^4)
          +a^5 (6 b^5+45 b^4 c+41 b^3 c^2+41 b^2 c^3+45 b c^4+6 c^5)
           +2 a^4 b c (15 b^4+94 b^3 c-14 b^2 c^2+94 b c^3+15 c^4)
           -6 a^3 (b^7+10 b^6 c-11 b^4 c^3-11 b^3 c^4+10 b c^6+c^7)
           +a^2 (b^2-c^2)^2 (2 b^4-19 b^3 c-158 b^2 c^2-19 b c^3+2 c^4)
           +a (b-c)^4 (b+c)^3 (2 b^2+27 b c+2 c^2)
            -(b-c)^6 (b+c)^4 : .... : ....).
 
  (6 - 9 - 13) - search numbers  of W: (0.777426283641197, 0.0129765810797222, 3.27286856409479)
  
 
 *** 2. ABC, A*B*C* are orthologic.
 
The orthologic center (ABC, A*B*C) is  X(5557)=Garcia-Feuerbach Point GF(1/3),   where GF is the mapping defined at X(5550).     (Emmanuel José Garcia; September 11, 2013).

The orthologic center (A*B*C*, ABC) is:

V =  2(r+10R) X(5) - 5R X(40).

V = (a^5 (b^2-10 b c+c^2)
         -a^4 (b^3+11 b^2 c+11 b c^2+c^3)
        -2 a^3 (b-c)^2 (b^2+6 b c+c^2)
       +2 a^2 (b-c)^2 (b^3+7 b^2 c+7 b c^2+c^3)
       +a (b^2-c^2)^2 (b^2+18 b c+c^2)
        -(b-c)^4 (b+c)^3 : ... : ....).
          
 V = X(5)X(40) /\ X(946)X(12680)
         
  (6 - 9 - 13) - search numbers  of  V: (-3.10967432619234, -3.68825662960864, 7.62930722218683)
         
  Angel Montesdeoca