HYACINTHOS 26831

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle, P a point and A'B'C' the cevian triangle of Ο,

Denote:

Ma, Mb, Mc = the midpoints of AA', BB', CC', resp.

 

Mab, Mac = the orthogonal projections of Ma on BB', CC', resp.
Mbc, Mba = the orthogonal projections of Mb on CC', AA', resp.
Mca, Mcb = the orthogonal projections of Mc on AA', BB', resp.

R1, R2, R3 = the Euler lines of MaMabMac, MbMbaMbc, McMcaMcb, resp.

 

1. R1, R2, R3 are concurrent.
2. The parallels to R1, R2, R3 through Ma, Mb, Mc, resp. are concurrent.
3. The reflections of R1, R2, R3 in BC, CA, AB, resp. are concurrent (parallels)

[Angel Montesdeoca]:

 1. R1, R2, R3 are concurrent at W1 =  X(111)X(930)/\X(125)X(128)

W1 = (b^2-c^2)^4 (b^4+c^4)-5 (b^2-c^2)^2 (b^6+c^6) a^2+2 (5 b^8-3 b^6 c^2+2 b^4 c^4-3 b^2 c^6+5 c^8) a^4+(-12 b^6-5 b^4 c^2-5 b^2 c^4-12 c^6) a^6+(11 b^4+12 b^2 c^2+11 c^4)a^8-7 (b^2+c^2)a^10+2a^12 : ....: ...

 (6 - 9 - 13) - search numbers  of W1:  0.847648488159698, 7.42063572482228, -1.88792109135090
 
   2. The parallels to R1, R2, R3 through Ma, Mb, Mc, resp. are concurrent at W2 = X(5)X(49)/\X(115)X(128)
  
   W2 =  (b^2-c^2)^6 (b^4+c^4)-5 (b^2-c^2)^4 (b^6+c^6) a^2+(b^2-c^2)^2 (11 b^8-2 b^6 c^2+2 b^4 c^4-2 b^2 c^6+11 c^8) a^4+(-15 b^10+11 b^8 c^2-2 b^6 c^4-2 b^4 c^6+11 b^2 c^8-15 c^10) a^6+(15 b^8+2 b^6 c^2+2 b^2 c^6+15 c^8) a^8+(-11 b^6-6 b^4 c^2-6 b^2 c^4-11 c^6)a^10+(5 b^4+4 b^2 c^2+5 c^4) a^12+(-b^2-c^2) a^14 : .... : ....
  
    (6 - 9 - 13) - search numbers  of W2: -5.08706591787695, 8.94632052291315, -0.205065456473827
   
        3. The reflections of R1, R2, R3 in BC, CA, AB, resp. are concurrent (parallels) at X(1154), the point where the Euler line of the orthic triangle meets the line at infinity.
       
        Angel Montesdeoca