HYACINTHOS 26178

[Antreas P. Hatzipolakis]:

 

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

Denote:

 

A", B", C" = the reflections of A', B', C' in A, B, C, resp.   

N1, N2, N3 = the NPC centers of HB"C", HC"A", HA"B", resp.

Na, Nb, Nc = the reflections of N1, N2, N3 in AA', BB', CC', resp.

 

1. The orthocenter of N1N2N3 is the O.

2. ABC, N1N2N3 are orthologic.

The orthologic center (ABC, N1N2N3) is X74
The other one?

3. The circumcenter of NaNbNc is the midpoint of NH

4. ABC, NaNbNc are orthologic.
Orthologic centers?

 

 

[Angel Montesdeoca]:

 

***  2.    The orthologic center (N1N2N3, ABC) is X(5907)  =  COMPLEMENT OF X(185)

**** 4. 
=== The orthologic center (ABC, NaNbNc) is  the reflection U of X(4) in X(11792)

[   For infomation to X(11792) see Antreas Hatzipolakis and Angel Montesdeoca Hyacinthos 25266
 

Let ABC be a triangle. Denote:
Labc = the reflection of AB in AC
Ca = the orthogonal projection of C on Labc
Lacb = the reflection of AC in AB
Ba = the orthogonal projection of B on Lacb
Similarly: Ab, Cb,  Bc, Ac
Ma, Mb, Mc = the midpoints of BaCa, CbAb, AcBc, resp.
The NPCs of ABC, MaBC, MbCA, McAB are concurrent at X(11792)   ]
        
           U = (a^16-b^2 c^2 (b^2-c^2)^6
                -6 a^14 (b^2+c^2)
                +2 a^12 (8 b^4+7 b^2 c^2+8 c^4)
                -5 a^10 (5 b^6+b^4 c^2+b^2 c^4+5 c^6)
                +a^8 (25 b^8-18 b^6 c^2+11 b^4 c^4-18 b^2 c^6+25 c^8)
                -a^6 (b^2-c^2)^2 (16 b^6-3 b^4 c^2-3 b^2 c^4+16 c^6)
                +a^4 (b^2-c^2)^2 (6 b^8-19 b^6 c^2-23 b^4 c^4-19 b^2 c^6+6 c^8)
                -a^2 (b^2-c^2)^4 (b^6-8 b^4 c^2-8 b^2 c^4+c^6) : ... : ....),           
  
U lies on  the circumcircle and  the lines X(i)X(j) for these {i, j}: {4,11792}, {30,11703}, {99,1232}, {110,140}, {112,6748}, {476,5899}, {953,5957}, {2687,5959}, {2699,5958}.

  with (6 - 9 - 13) - search numbers (11.4518521443969,  13.1930353900201,  -10.7784456239820).        


=== The orthologic center (NaNbNc, ABC) is 

       V = (a^2 (a^6 (b^2+c^2)                    
                     -a^4 (3 b^4+8 b^2 c^2+3 c^4)
                     +a^2 (3 b^6+b^4 c^2+b^2 c^4+3 c^6)
                     -(b^2-c^2)^2 (b^4-4 b^2 c^2+c^4)) : ... : ...),
  

V is the midpoint of X(i) and X(j), for these {i, j}: {52,382}, {185,3146}, {3627,10263}, {5073,10575}, {5889,11381}, {6243,12162}, {10733,13417}.

V is the reflection of X(i) in X(j), for these {i, j}:  {3,10110}, {20,9729}, {186,13446}, {389,5446}, {548,10095}, {550,5462}, {1071,12109}, {1216,546}, {1350,9822}, {2071,13376}, {5462,12002}, {5907,4}, {10112,13142}, {10575,13382}, {10625,11793}, {10627,3850}, {11573,5806}, {11574,5480}, {11591,3861}, {12103,12006}, {13474,3627}.

V lies on the circumcircle of NaNbNc and  the lines X(i)X(j) for these {i, j}: {2, 13348}, {3, 5943}, {4, 69}, {5, 3819}, {20, 51}, {22, 11424}, {23, 13367}, {25, 13346}, {26, 11430}, {30, 143}, {52, 382}, {182, 10790}, {185, 3060}, {186, 13446}, {343, 1907}, {373, 3523}, {376, 9781}, {381, 10625}, {394, 5198}, {517, 12527}, {546, 1216}, {548, 5892}, {550, 5462}, {568, 5073}, {569, 12083}, {575, 10984}, {576, 1181}, {578, 7387}, {631, 6688}, {970, 1012}, {1071, 12109}, {1092, 10594}, {1147, 7530}, {1154, 3853}, {1199, 8718}, {1350, 9822}, {1351, 1498}, {1503, 10112}, {1539, 13433}, {1598, 9306}, {1629, 1941}, {1657, 9730}, {2071, 13376}, {2393, 2883}, {2777, 11800}, {2979, 3832}, {3091, 3917}, {3098, 7395}, {3522, 5640}, {3525, 10219}, {3529, 3567}, {3530, 13364}, {3533, 12045}, {3543, 5889}, {3627, 10263}, {3830, 6243}, {3839, 11444}, {3843, 5891}, {3845, 6101}, {3850, 10170}, {3851, 13340}, {3855, 7999}, {3861, 11591}, {5012, 12087}, {5056, 5650}, {5059, 10574}, {5068, 7998}, {5092, 10323}, {5097, 7592}, {5480, 6823}, {5806, 11573}, {7517, 10282}, {8549, 9914}, {8681, 11477}, {9714, 11202}, {9815, 10996}, {9909, 11425}, {9969, 12362}, {10299, 11465}, {10601, 13347}, {10628, 12295}, {10733, 13417}, {11438, 12085}, {11807, 11819}, {12006, 12103}.

   with (6 - 9 - 13) - search numbers (-9.43973422559837, -9.39477334459159, 14.5015387476701).

 

 

Angel  Montesdeoca