HYACINTHOS 25041

[Antreas P. Hatzipolakis]:

 

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

Denote:

Ab, Ac = the orthogonal projections of A' on BH, CH, resp.

N1 = the NPC center of A'AbAc.

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.

The redical center of the circles (A', A'Na), (B', B'Nb), (C', C'Nc) lies on the OI line.

[Angel Montesdeoca]:


The redical center of the circles (A', A'Na), (B', B'Nb), (C', C'Nc) is

 W = (R+4r) X[1] - R X[3],

Barycentric coodinates:

(a (-2 a^6 + 4 a^5 (b + c) + 2 a^4 (b^2 - 6 b c + c^2) + a^3 (-8 b^3 + 7 b^2 c + 7 b c^2 - 8 c^3)+ 2 a^2 (b^4 + 4 b^3 c - 9 b^2 c^2 + 4 b c^3 + c^4) + a (b - c)^2 (4 b^3 - 3 b^2 c - 3 b c^2 + 4 c^3) - 2 (b - c)^4 (b + c)^2) : ... : ...)

    with (6-9-13)-search numbers (-3.89503864380390, -2.92165864199079, 7.46106137734903).
    W lies on lines: {1,3}, {3244,10265}, {3822,5901}, {5141,5886}
   
    Angel Montesdeoca