HYACINTHOS 24935

 

[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 HB, HC, resp.

Na = the NPC center of A'AbAc

N1 = the reflection of Na n BC.

Similarly N2, N3.

The circumcenter of N1N2N3 lies on the Euler line of ABC.



[Angel Montesdeoca]:

*** The circumcenter of N1N2N3  is X(3628) is the centroid of the set {A', B', C', X(5)}, where A'B'C' is the medial triangle.

[APH]:

 

Thanks, Angel !

And the radical center of the circles (N1), (N2), (N3) [ = reflections of the NPCs of A'AbAc, B'BcBa, C'CaCb in BC, CA, AB, resp.] lies on the Euler line.

[Angel Montesdeoca]:

The radical center of the circles (N1), (N2), (N3) [ = reflections of the NPCs of A'AbAc, B'BcBa, C'CaCb in BC, CA, AB, resp.] is W the midpoint of  points on Euler line X(5133)=inverse-in-nine-point- circle of  X(23) and  X(7499) = {X(2),X(22)}-harmonic conjugate of X(5).

W = (2 a^6-3 a^4 (b^2+c^2)-2 a^2 (b^4+4 b^2 c^2+c^4)+3 (b^2-c^2)^2 (b^2+c^2): ...: ... ),

with (6,9,13) search numbers (1.40233422648944,  0.529114910493857,  2.62712297780120)

W lies on lines: {2,3}, {230,233}, {232,10184}, {590,8281}, {615,8280}, {3793,8878}, {3819,9969}, {6723,10219}.

Angel Montesdeoca