HYACINTHOS 25831

 [Antreas P. Hatzipolakis]:

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

Denote:

T1, T2, T3 = the pedal triangles of A, B, C wrt triangle A'B'C'.

Na, Nb, Nc = the NPC centers of T1, T2, T3, resp.

3. ABC, NaNbNc are orthologic. Orthologic centers?

 

[Angel Montesdeoca]¨

**** Orthologic center (ABC, NaNbNc) is X(4).
Orthologic center (NaNbNc,ABC) is

V = ( a (a^5 (b+c)-a^4 (b^2+c^2)-2 a^3 (b^3+b^2 c+b c^2+c^3)+2 a^2 (b-c)^2 (b^2+c^2)-(b^2-c^2)^2 (b^2-4 b c+c^2)+a (b-c)^2 (b+c)^3) : .. : ...),

V is the midpoint of X(i) and X(j) for these {i,j}: {1,7686}, {4,12675}, {942,946}, {1482,5836}, {3874,5777}, {5045,5806}, {5173,7680}, {5572,5805}, {5884,9856}, {6583,9955}, {7682,12915}, {9943,12699}.

V is the  reflection of X(4662) in X(9956).

V lies on lines:  {1,227}, {3,3742}, {4,354}, {5,518}, {7,10309}, {10,12864}, {11,12691}, {40,5439}, {57,11496}, {65,3086}, {72,5231,8227}, {84,10980}, {140,517}, {210,3090}, {226,7681}, {388,5804},{392,9624}, {405,12704},{496,942}, {497,12710}, ....

with (6,9,13)- search numbers (-0.441569019626854, -0.604640449545626, 4.26306280218990).

***  ABC and  NaNbNc are homothetic. Homothetic center: X(354)

**** A'B'C', NaNbNc are orthologic.

Orthologic center (A'B'C', NaNbNc) is X(1).
Orthologic center (NaNbNc,ABC) is  X(5045) = (r+4R) X(1) - r X(3)

Angel Montesdeoca