HYACINTHOS 26987

[Antreas P. Hatzipolakis]:

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

Denote:

A"B"C" = the midheight triangle 
(ie A", B", C" = the midpoints of AA', BB', CC', resp.)

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

Ab, Ac = the orthogonal projections of A* on AB, AC, resp.
Bc, Ba = the orthogonal projections of B* on BC, BA, resp.

Ca, Cb = the orthogonal projections of C* on CA, ab, resp.

 

A*B*C* = the triangle bounded by AbAc, BcBa, CaCb

 

M1, M2, M3 = the midpoints of AbAc, BcBa, CaCb, resp.

 

 

1. A'B'C', A*B*C* are homothetic.

 

2. The perpendicular bisectors of AbAc, BcBa, CaCb are concurrent

 

3. ABC, M1M2M3 are orthologic

 

4. A'B'C', M1M2M3 are orthologic

 

5. A*B*C*, M1M2M3 are orthologic.

 

 

 

[Angel Montesdeoca]:

 1. A'B'C', A*B*C* are homothetic with homothetic center X(51).

    2. The perpendicular bisectors of AbAc, BcBa, CaCb are concurrent at X(5907).

    3. ABC, M1M2M3 are orthologic
   
   ***   The orthology center of ABC with respect to M1M2M3 is 

Z3 with first barycentric coordinate:

    
 a^10+3 a^8 (b^2+c^2)-14 a^6 (b^2-c^2)^2+2 a^4 (7 b^6-15 b^4 c^2-15 b^2 c^4+7 c^6)-a^2 (b^2-c^2)^2 (3 b^4+10 b^2 c^2+3 c^4)-(b^2-c^2)^4 (b^2+c^2) ::

Z3 lies on lines X(i)X(j) for these {i, j}: {2, 64}, {3, 11821}, {4, 5943}, {5, 3426}, {6, 20}, {30, 3527}, {54, 376}, {65, 497}, {66, 6815}, {69, 185}, {73, 1040}, {74, 631}, {285, 6904}, {382, 3531}, {1105, 1249}, {1173, 3529}, {1192, 10565}, {1204, 7494}, {1243, 6851}, {1245, 2999}, {1370, 14542}, {1593, 3618}, {1899, 15077}, {3431, 3528}, {3519, 11411}, {3522, 3796}, {3523, 3532}, {3524, 11270}, {3525, 13452}, {3537, 5562}, {3575, 14927}, {3832, 14490}, {3855, 13603}, {4846, 6643}, {5067, 11738}, {5085, 5894}, {5486, 5889}, {5663, 11487}, {5878, 6804}, {5900, 15102}, {6000, 6803}, {6391, 6776}, {6639, 11559}, {6997, 12279}, {7392,11381}, {7395, 12250}, {7400, 10605}, {7401, 10575}, {7544, 15321}, {8814, 10446}, {10574, 11433}, {11413, 11427}, {11744, 13203}, {12174, 14826}, {14944, 15005}.

Z3 = complement of X(11469),

Z3 is the reflection of X(i) in X(j), for these {i, j}: {4,9815}, {11821,3}.

 (6 - 9 - 13) - search numbers  of Z3: (10.2893367499920, 10.0640546688309, -8.07568263497177)
 
 
    ***  The orthology center of M1M2M3 with respect to ABC is X(13568).
   
     
    4. A'B'C', M1M2M3 are orthologic
   
    ***  The orthology center of A'B'C' with respect to M1M2M3 is 

Z4 with first barycentric coordinate:
    
5 a^10 + a^8(b^2+c^2)-2 a^6 (15 b^4+2 b^2 c^2+15 c^4)+2 a^4 (17 b^6-9 b^4 c^2-9 b^2 c^4+17 c^6) -a^2 (b^2-c^2)^2 (7 b^4+2 b^2 c^2+7 c^4) -3 (b^2-c^2)^4 (b^2+c^2) ::
 

Z4 lies on lines X(i)X(j) for these {i, j}: {2, 3574}, {4, 11469}, {6, 20}, {113, 3089}, {155, 7487}, {185, 3060}, {193, 3575}, {1829, 12528}, {1843, 5889}, {3088, 7689}, {3448, 13202}, {3543, 5895}, {6193, 13431}, {7396, 9786}, {7398, 11745}, {7408, 12111}, {10565, 12233}.

Z4 = anticomplement of X(11821)

Z4 is the reflection of X(i) in X(j), for these {i, j}: {11469,4}, {11821,9815}

 (6 - 9 - 13) - search numbers  of Z4: (1.33732830066479, 1.80704321571357, 1.77240611687585)
  
   ***     The orthology center of M1M2M3 with respect to A'B'C' is X(5907)
       
   
    5. A*B*C*, M1M2M3 are orthologic.
   
   ***  The orthology center of A*B*C* with respect to M1M2M3 is 

Z5 with first barycentric coordinate:
    
 a^2 (-a^12 (b^2+c^2)
       +a^10 (4 b^4-46 b^2 c^2+4 c^4)
       +a^8 (-5 b^6+11 b^4 c^2+11 b^2 c^4-5 c^6)
       +5 a^4 (b^10-55 b^8 c^2+22 b^6 c^4+22 b^4 c^6-55 b^2 c^8+c^10)      
       +4 a^6 (57 b^6 c^2-2 b^4 c^4+57 b^2 c^6)
       -2 a^2 (b^2-c^2)^2 (2 b^8-25 b^6 c^2-34 b^4 c^4-25 b^2 c^6+2 c^8)
       +(b^2-c^2)^4 (b^6+29 b^4 c^2+29 b^2 c^4+c^6))

 (6 - 9 - 13) - search numbers  of Z5: (3.16238237760072,  4.75184832290143, -1.10863776207079)
  
     
    *** The orthology center of M1M2M3 with respect to A*B*C* is  X(5907).
   
    Angel Montesdeoca