HYACINTHOS 25862

 [Antreas P. Hatzipolakis]:

 

Let ABC be a triangle and A'B'C' the cevian triangle of G.

    Denote:

    A2 = the orthogonal projection of A' on BB'
    A23 = the orthogonal projection of A2 on CC'

    A3 = the orthogonal projection of A' on CC'
    A32 = the orthogonal projection of A3 on BB'

    A23A32 =: La. Similarly Lb, Lc

    Aa,Ab, Ac = the orthogonal projections of A on La, Lb, Lc, resp.
    Ba,Bb, Bc = the orthogonal projections of B on La, Lb, Lc, resp.
    Ca,Cb, Cc = the orthogonal projections of C on La, Lb, Lc, resp.

    Ga, Gb, Gc = the centroids of AaAbAc, BaBbBc, CaCbCc, resp.

    1. The centroid of GaGbGc is the G.

    2. ABC, GaGbGc are orthologic.

 

[Angel Montesdeoca]:


*** The  orthology center of ABC with respect to GaGbGc is U  =   2 X(2) - X(6322) 

 U = (4 a^8+10 a^6 (b^2+c^2)+33 a^4 b^2 c^2+a^2 (4 b^6+6 b^4 c^2+6 b^2 c^4+4 c^6)-4 (2 b^8+2 b^6 c^2-9 b^4 c^4+2 b^2 c^6+2 c^8) : ... : ...),
 
 U is reflection of X(6322) in X(2)
 U lies on the lines: {2,6322}, {381,8704}, {542,6232}, {599,3734}, {5094,10162}, {6032,11163},{7840,9464}, {9829,10130}.

with (6,9,13)-search numbers (-1.62225895713714, -1.17105826632023, 5.20013203111553).
 
 The  orthology center of GaGbGc with respect to ABC  is  V  =  X(10166) - 2 X(10173)
 
 V = (4 a^6+9 a^4 (b^2+c^2)+a^2 (9 b^4+6 b^2 c^2+9 c^4)-2 (7 b^6-9 b^4 c^2-9 b^2 c^4+7 c^6) : ... : ...),
 
  V is reflection of X(6322) in X(2)
 V lies on the lines: {2,1495}, {125,3363}, {9830,10162}, {10166,10173}.

 with (6,9,13)-search numbers (0.533150296499902, -0.602859495229007, 3.81195938021989).
 
 Angel Montesdeoca