HYACINTHOS 25833

 [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'.

Oa, Ob, Oc = the circumcenters of T1, T2, T3, resp.

1. ABC, OaObOc are orthologic
2. A'B'C', OaObOc are orthologic

Orthologic centers?

 

[Angel Montesdeoca]:



**** 1.   Orthologic center (ABC, OaObOc) is X(1389)

         Orthologic center ( OaObOc, ABC) is X(7686)
        
 **** 2.   Orthologic center (A'B'C', OaObOc) is Z=X(5559) +X(5903)  
 
 Z   = (a (a^5 (b+c)-(b^2-c^2)^2 (b^2-3 b c+c^2)-a^4 (b^2+4 b c+c^2)+a^3 (-2 b^3+3 b^2 c+3 b c^2-2 c^3)+a (b-c)^2 (b^3-2 b^2 c-2 b c^2+c^3)+a^2 (2 b^4+b^3 c-12 b^2 c^2+b c^3+2 c^4))   :  ....   : ...),
 
 Z  is the midpoint of X(5559) and X(5903) .

 with (6,9,13)- search numbers (1.05135765621466, 1.02167741998769, 2.44810734981690)

 
    Orthologic center ( OaObOc, A'B'C') is X(9957).
   
    Angel Montesdeoca