HYACINTHOS 25310

 

[Antreas P. Hatzipolakis]:

 

 

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

Denote:

Nab, Nac = the NPC centers of BB'A', CC'A', resp.

Nbc, Nba = the NPC centers of CC'B', AA'B', resp.

Nca, Ncb = the NPCs of AA'C', BB'C', resp.

A1B1C1 = the triangle bounded by NabNac, NbcNba, NcaNcb

A2B2C2 = the triangle bounded by NbaNca, NcbNab, NacNbc

A3B3C3 = the triangle bounded by NbcNcb, NcaNac, NabNba

I think the following are true:

1. ABC, A1B1C1 are parallelogic.

The parallelogic center (ABC, A1B1C1) is X110 [Euler line reflection point]

2. A'B'C', A1B1C1 are orthologic.

3. ABC, A2B2C2 are bilogic:
3.1. parspective.

3.2. orthologic

4. A'B'C', A2B2C2 are orthologic.

5. ABC, A3B3C3 are perspective (homothetic)

6. A'B'C', A3B3C3 are orthologic.

7. A1B1C1, A2B2C2 are orthologic.

8. A1B1C1, A3B3C3 are orthologic.

9. A2B2C2, A3B3C3 are orthologic

[Angel Montesdeoca]:

1. ABC, A1B1C1 are parallelogic.
The parallelogic center (ABC, A1B1C1) is X110 [Euler line reflection point]
The parallelogic center ( A1B1C1, ABC) is 
W12 = (a^2 (-a^8 (b^2+c^2) 
            +2 a^6 (b^4+b^2 c^2+c^4)
            -3 a^4 b^2 c^2 (b^2+c^2)
            +a^2 (-2 b^8+5 b^6 c^2-4 b^4 c^4+5 b^2 c^6-2 c^8)
            +(b^2-c^2)^4 (b^2+c^2) ) : ... : ... ),
with (6-9-13)-search numbers (-0.272843112989601,  0.216129312553261,   3.61696408690424).
W12 is the midpoint of X(i) and X(j) for these {i,j}: {52,265}, {185,10733}, {895,1843}, {10263,10264}.

2. A'B'C', A1B1C1 are orthologic.
The orthologic center (A'B'C', A1B1C1) is  X(1986) = Hatzipolakis Refletion Point:
Let A'B'C' be the orthic triangle of triangle ABC. Let Ab be the reflection of A in C', and define Ac, Bc, Ba, Ca, Cb functionally. Then the nine-point circles of the triangles AAbAc,    BBcBa,    CCaCb,   concur in X(1986).

The orthologic center (A1B1C1, A'B'C') is
  W22 = ( -2 a^10
          +3 a^8 (b^2+c^2)
          -6 a^6 b^2 c^2
          +a^4 (2 b^6-b^4 c^2-b^2 c^4+2 c^6)
         -a^2 (b^2-c^2)^2 (6 b^4-b^2 c^2+6 c^4)
          +3 (b^2-c^2)^4 (b^2+c^2) : ... : ... ),
with (6-9-13)-search numbers (0.375526125401512, -0.0267904101447112, 3.485891938745402).
W32 is the midpoint of X(i) and X(j) for these {i,j}:   {4, 10264}, {5, 265}, {74, 3627}, {125, 10113}, {550, 10733}, {3845, 9140}.


3. ABC, A2B2C2 are bilogic:
3.1. ABC, A2B2C2 are not perspective.
3.2.  The orthologic center (ABC, A2B2C2) is  X(54) =  Kosnita Point.
         The orthologic center (A2B2C2, ABC) is  
         W32 = ( a^2 (a^12 (b^2+c^2)
                      -2 a^10 (2 b^4+b^2 c^2+2 c^4)
                      +5 a^8 (b^6+c^6)-(b^2-c^2)^4 (b^6+b^4 c^2+b^2 c^4+c^6)
                      -a^6 b^2 c^2 (b^4-4 b^2 c^2+c^4)
                     +a^4 (-5 b^10+8 b^8 c^2+3 b^6 c^4+3 b^4 c^6+8 b^2 c^8-5 c^10)
                      +a^2 (b^2-c^2)^4 (4 b^4+7 b^2 c^2+4 c^4) ) : ... : ...),
with (6-9-13)-search numbers (9.50430905285704, 0.601631864336333, -1.16245406395172).
W32 is the midpoint of X(i) and X(j) for these {i,j}:    {52,7691}, {185,6288}.

4. A'B'C', A2B2C2 are orthologic.
     The orthologic center (A'B'C', A2B2C2) is   X(6152) = ORTHIC-TRIANGLE-ORTHOLOGIC CENTER OF REFLECTION TRIANGLE.
       The orthologic center (A2B2C2, A'B'C') is   W42 = X(4) + 3 X(195)
       W42 = (2 a^10
                  -11 a^8 (b^2+c^2)
                  +2 a^6 (10 b^4+7 b^2 c^2+10 c^4)
                  +a^4 (-14 b^6+5 b^4 c^2+5 b^2 c^4-14 c^6)
                  +a^2 (b^2-c^2)^2 (2 b^4-b^2 c^2+2 c^4)
                  +(b^2-c^2)^4 (b^2+c^2) : ... : ...),
with (6-9-13)-search numbers (-15.7875656020820,  10.0775520867938,  3.95046639201115).
    
5. ABC, A3B3C3 are perspective (homothetic). The homothetic center is X(51) = Centroid of orthic triangle.

6. A'B'C', A3B3C3 are orthologic.
    The orthologic center (A'B'C', A3B3C3) is  X(4) = Orthocenter.
    The orthologic center (A3B3C3, A'B'C') is  X(5446) = midpoint of X(4) and X(52).  X(52) =Orthocenter of orthic triangle.
   
7. A1B1C1, A2B2C2 are orthologic.
   The orthologic center (A1B1C1, A2B2C2) is    W71 = X(54) + X(265).
   W71 = ( -a^14 (b^2+c^2)
               +3 a^12 (b^2+c^2)^2
                -a^10 (b^6+11 b^4 c^2+11 b^2 c^4+c^6)
                 +a^8 (-5 b^8+12 b^6 c^2+6 b^4 c^4+12 b^2 c^6-5 c^8)
                 +a^6 (5 b^10-11 b^8 c^2+3 b^6 c^4+3 b^4 c^6-11 b^2 c^8+5 c^10)
                 +a^4 (b^2-c^2)^2 (b^8+4 b^6 c^2+5 b^4 c^4+4 b^2 c^6+c^8)
                 -a^2 (b^2-c^2)^4 (3 b^6+5 b^4 c^2+5 b^2 c^4+3 c^6)
                +(b^2-c^2)^6 (b^2+c^2)^2 : ... : ... ),
 with (6-9-13)-search numbers (-2.99478927165030, 5.50944837494833, 1.20864142462798).
       The orthologic center (A2B2C2, A1B1C1) is     W72 = X(54) + X(7728).
   W72 = (-3 a^14 (b^2+c^2)
               +a^12 (11 b^4+6 b^2 c^2+11 c^4)
                -a^10 (11 b^6+5 b^4 c^2+5 b^2 c^4+11 c^6)
               +a^8 (-5 b^8+14 b^6 c^2-14 b^4 c^4+14 b^2 c^6-5 c^8)
               +a^6 (15 b^10-23 b^8 c^2+11 b^6 c^4+11 b^4 c^6-23 b^2 c^8+15 c^10)
               -a^4 (b^2-c^2)^2 (7 b^8+6 b^6 c^2-3 b^4 c^4+6 b^2 c^6+7 c^8)
               -a^2 (b^2-c^2)^4 (b^6-3 b^4 c^2-3 b^2 c^4+c^6)
               +(b^2-c^2)^6 (b^2+c^2)^2  : ... : ... ),
 with (6-9-13)-search numbers (-8.86958919174496, -1.28624174412460,  8.62479608556831).      
        
8. A1B1C1, A3B3C3 are orthologic.
      The orthologic center (A1B1C1, A3B3C3) is  
         W81 = ( a^2 (a^12 (b^2+c^2)
                     -2 a^10 (2 b^4+b^2 c^2+2 c^4)
                     +a^8 (5 b^6+4 b^4 c^2+4 b^2 c^4+5 c^6)
                    +a^6 (-13 b^6 c^2+12 b^4 c^4-13 b^2 c^6)
                    -a^4 (5 b^10-20 b^8 c^2+13 b^6 c^4+13 b^4 c^6-20 b^2 c^8+5 c^10)
                    +a^2 (b^2-c^2)^2 (4 b^8-5 b^6 c^2-10 b^4 c^4-5 b^2 c^6+4 c^8)
                    -(b^2-c^2)^4 (b^6+b^4 c^2+b^2 c^4+c^6) ) : ... : ...),
with (6-9-13)-search numbers (5.95664803957184, 6.42918691016165, -3.55953324339146).
W81 is the midpoint of X(i) and X(j) for these {i,j}:    {52,74}, {185,265}, {3448,11562}, {6102,10264}, {10575,10733}.
    The orthologic center (A3B3C3, A1B1C1) is  
         W82 = (a^2 (a^12 (b^2+c^2)
                   -2 a^10 (2 b^4+3 b^2 c^2+2 c^4)
                    +a^8 (5 b^6+6 b^4 c^2+6 b^2 c^4+5 c^6)
                    +a^6 (3 b^6 c^2-16 b^4 c^4+3 b^2 c^6)
                    -5 a^4 (b^10-b^6 c^4-b^4 c^6+c^10)
                    +a^2 (b^2-c^2)^2 (4 b^8-b^6 c^2+12 b^4 c^4-b^2 c^6+4 c^8)
                     -(b^2-c^2)^6 (b^2+c^2)) : ... : ...),
with (6-9-13)-search numbers (-6.14764303308426, -6.57956080651967, 11.0331187478446).
W82 is the midpoint of X(i) and X(j) for these {i,j}:   {52,7728}, {185,10721}, {382,11562}, {1843,10752}, {7722,11381}.

9. A2B2C2, A3B3C3 are orthologic

The orthologic center (A2B2C2, A3B3C3) is  
         W91 = ( a^2 (-a^12 (b^2+c^2)
                       +2 a^10 (2 b^4+b^2 c^2+2 c^4)
                       -5 a^8 (b^6+c^6)+(b^2-c^2)^4 (b^6+b^4 c^2+b^2 c^4+c^6)
                       +a^6 b^2 c^2 (b^4-4 b^2 c^2+c^4)
                       +a^4 (5 b^10-8 b^8 c^2-3 b^6 c^4-3 b^4 c^6-8 b^2 c^8+5 c^10)
                       -a^2 (b^2-c^2)^4 (4 b^4+7 b^2 c^2+4 c^4) ): ... : ...),
with (6-9-13)-search numbers (9.50430905285704,  0.601631864336333,  -1.16245406395172).
W91 is the midpoint of X(i) and X(j) for these {i,j}:    {52,7691},  {185,6288}.
    The orthologic center (A3B3C3, A2B2C2) is  
         W92 = (a^2 (a^12 (b^2+c^2)
                     -2 a^10 (2 b^4+3 b^2 c^2+2 c^4)
                     +a^8 (5 b^6+6 b^4 c^2+6 b^2 c^4+5 c^6)
                     +3 a^6 b^2 c^2 (b^4+c^4)
                     -5 a^4 (b^10-b^6 c^4-b^4 c^6+c^10)
                     +a^2 (b^2-c^2)^2 (4 b^8-b^6 c^2-4 b^4 c^4-b^2 c^6+4 c^8)
                     -(b^2-c^2)^6 (b^2+c^2) ) : ... : ...),
with (6-9-13)-search numbers (3.27481790029560,  -5.61142573327205, 6.01404326634398).
W92 is the midpoint of X(i) and X(j) for these {i,j}:  {52,6288}, {973,11576}, {3574,6152}, {5446,6153}.

Angel Montesdeoca