HYACINTHOS 23990

-

Antreas P. Hatzipolakis

 

[APH]:

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

    Denote:

    H' = the orthocenter of A'B'C'

    Ab, Ac = the orth. projections of A on H'B', H'C', resp.

    L1 = te Euler line of AAbAc. Similarly L2, L3

    L1, L2, L3 are concurrent.

    Furthermore:

The parallels to L1, L2, L3 through

1. A, B,C, resp.

2. A', B', C', resp.

3. A", B", C" = vertices of the orthic triangle of A'B'C'

are concurrent.

***   L1, L2, L3 are concurrent in X(5836) = midpoint of X(8) and X(65).
***   The parallels to L1, L2, L3 through  A, B, C, resp. are concurrent in X(8).
***   The parallels to L1, L2, L3 through  A', B', C', resp. are concurrent in X(145).
***   The parallels to L1, L2, L3 through  A'', B'', C'', resp. are concurrent in
   W=(2 a^4-a^3 (b+c)+a (b-c)^2 (b+c)-(b^2-c^2)^2-a^2 (b^2-6 b c+c^2):...:...)
  
   W=(2R-r)X(1)+rX(4)
   ETC numbers search  (0.850772564114176,  0.753388534762742,  2.72642354363440)

Angel Montesdeoca