HYACINTHOS 25817

 [Antreas P. Hatzipolakis]:

 

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

    Denote:

    Aa, Ab, Ac = the reflections of A in B'C', C'A', A'B', resp.
    Ba, Bb, Bc = the reflections of B in B'C', C'A', A'B', resp.
    Ca, Cb, Cc = the reflections of C in B'C', C'A', A'B', resp.

    Oa, Ob, Oc = the circumcenters of AaAbAc, BaBbBc, CaCbCc, resp.

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

 

[Angel Montesdeoca]:


*** The circumcenter of OaObOc is 

W = (a^8 (b^2+c^2)-2 a^6 (b^2+c^2)^2+2 a^4 b^2 c^2 (b^2+c^2)+2 a^2 (b^8-b^6 c^2-b^2 c^6+c^8) -(b^2-c^2)^4 (b^2+c^2): .... : ....),

W is the  midpoint of X(4) and X(12084). 
W is the  reflection of X(i) in X(j) for these {i,j}:  {5,10224}, {26,10020}, {156,9820}, {550,10226}, {1658,140}, {12107,10125}.
W lies on lines:  {2, 3}, {11, 8144}, {50, 9722}, {52, 125}, {70, 1993}, {113, 11381}, {141, 12061}, {155, 1853}, {156, 1503}, {343, 6101}, {496, 9630}, {511, 5449}, {524, 11255}, {590, 11265}, {615, 11266}, {1154, 12359}, {1209, 3917}, {1236, 3933}, {1568, 12162}, {1899, 12161}, {3574, 9730}, {3580, 6243}, {3925, 8141}, {5448, 6000}, {5480, 10095}, {5504, 6145}, {5663, 6247}, {7703, 11444}, {10539, 11550}, {11064, 12134}.
 
with (6-9-13) search numbers:  (5.00244120597201, 4.11972471686304, -1.52027164790789)
 
*** All triangles are related to one another by affine transformations. The fixed point of affine transformation (ABC, OaObOc) is X(125)= center of Jerabek hyperbola.

*** The triangles A'B'C' and OaObOc are perspective, with perspector X(141) = complement of X(6).

Angel Montesdeoca