HYACINTHOS 26203

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle.

Denote:

Na, Nb, Nc = the NPC centers of IBC, ICA, IAB, resp.

Nab, Nac = the reflections of A in Nb, Nc, resp.

(N1) = the NPC of ANabNac. Similarly (N2), (N3)

(N1), (N2), (N3) are concurrent at Y.

Equivalently:

Let A', B', C' be the reflections of A, B, C in NbNc, NcNa, NaNb, resp.

 

We have:

NPC of ANabNac = circumcircle of A'NbNc (by construction)
NPC of BNbcNba = circumcircle of B'NcNa
NPC of CNcaNcb = circumcircle of C'NaNb

The triangles NaNbNc, A'B'C' are circumcyclologic
(ie the circumcircles of NaNbNc, NaB'C', NbC'A', NcA'B' are concurrent at X

 the circumcircles of A'B'C', A'NbNc, B'NcNa, C'NaNb are concurrent at Y).

 

Also:

Let Ma, Mb, Mc be the midpoints of NabNac, NbcNba, NcaNcb, resp.

 

We have:

 

We have:

NPC of ANabNac = circumcircle of MaNbNc
NPC of BNbcNba = circumcircle of MbNcNa
NPC of CNcaNcb = circumcircle of McNaNb

The triangles NaNbNc, MaMbMc are circumcyclologic.
(ie the circumcircles of NaNbNc, NaMbMc, NbMcMa, NcMaMb are concurrent at Z

the circumcircles of MaMbMc, MaNbNc, MbNcNa, McNaNb are concurrent at Y ).

Which points are X,Y,Z ?

In general:

Let S1, S2, S3 be any points on (N1), (N2), (N3), resp.
The triangles NaNbNc, S1S2S3 are cyclologic.

[Angel Montesdeoca]:

[APH]:  Which points are X,Y,Z ?

**** X = (a^9 (b+c)    
      +a^8 (b^2-6 b c+c^2)
      +a^7 (-4 b^3+3 b^2 c+3 b c^2-4 c^3)
      -2 a^6 (b^4-3 b^3 c-3 b c^3+c^4)
      +a^5 (6 b^5-9 b^4 c+2 b^3 c^2+2 b^2 c^3-9 b c^4+6 c^5)
      +a^4 b c (4 b^4+b^3 c-14 b^2 c^2+b c^3+4 c^4)
      +a^3 (-4 b^7+3 b^6 c+2 b^4 c^3+2 b^3 c^4+3 b c^6-4 c^7)
      +a^2 (b^2-c^2)^2 (2 b^4-4 b^3 c+b^2 c^2-4 b c^3+2 c^4)
      +a (b-c)^2 (b+c)^3 (b^4+b^3 c-2 b^2 c^2+b c^3+c^4)
      -(b^2-c^2)^4 (b^2+c^2)       : ... : ...)

  
    X is the reflection of X(5620) in X (11)
    X= X(10)X(21)/\X(11)X(5620)
    (6 - 9 - 13) - search numbers of X: (2.71428567139167, -0.746776595932725, 2.90491643075703).
   

**** Y =  X(10)  - 2 X(125)

 Y = ((b+c) (-a^6
                     +a^4 (b^2+c^2)
                     -2 a^3 (b-c)^2 (b+c)
                    -a^2 (b^4-b^2 c^2+c^4)
                    +2 a (b^5-b^4 c-b c^4+c^5)
                    +(b^2-c^2)^2 (b^2+c^2) ) :...:...)   

Y is the complement of  X(2948)= EXCENTRAL-ISOGONAL CONJUGATE OF X(30)

Y is the midpoint of X(i) and X(j) for these {i, j}: {1,3448}, {944,12407}, {962,9904}, {7984,13211}, {10620,12699}

Y is the reflection of X(i) in X(j)  for these {i, j}:  {10,125}, {110,1125}, {946,12261}, {4297,11709}, {11699,5901}, {11720,11735}, {12778,6684}.

Y lies on lines X(i)X(j) for these {i, j}: {1, 3448}, {2, 2948}, {5, 2771}, {10, 125}, {67, 5847}, {74, 516}, {80, 5620}, {110, 1125}, {113, 3817}, {141, 2836}, {146, 1699}, {226, 3028}, {265, 515}, {399, 5886}, {517, 10264}, {519, 7984}, {542, 551}, {690, 4049}, {758, 10693}, {944, 12407}, {946, 5663}, {950, 12904}, {962, 9904}, {1511, 10165}, {2778, 6247}, {2784, 11005}, {2796, 11006}, {2842, 11814}, {3120, 6788}, {3576, 12383}, {4292, 10081}, {4297, 11709}, {4304, 12896}, {4466, 6326}, {5603, 12317}, {5901, 11699}, {6684, 12778}, {6699, 10164}, {10065, 10624}, {10088, 13411}, {10106, 12903}, {10272, 11230}, {10620, 12699}.

(6 - 9 - 13) - search numbers of Y: (3.88164446935990,.37526223955 876,1.17989143903010).

**** Z = X(946) =  ISOGONAL CONJUGATE OF X(944).  X(944) = HOFSTADTER TRAPEZOID POINT

Angel  Montesdeoca