HYACINTHOS 25814

[Antreas P. Hatzipolakis]:

https://beta.groups.yahoo.com/neo/groups/Hyacinthos/conversations/messages/25768

Let ABC be a triangle and HaHbHc, NaNbNc the pedal triangles of H, N, resp.

Denote:

A', B', C' = the reflections of N in BC, CA, AB, resp.

H1, H2, H3 = the midpoints of HA', HB', HC', resp.

O1, O2, O3 = the midpoints of OA', OB', OC', resp.

Nha, Nhb, Nhc = the NPC centers of H1O2O3, H2O3O1, H3O1O2, resp.
Noa, Nob, Noc = the NPC centers of O1H2H3, O2H3H1, O3H1H2, resp.

1. The lines NhaNoa, NhbNob, NhcNoc are concurrent on the Euler line of HaHbHc
2. The parallels to NhaNoa, NhbNob, NhcNoc through A, B, C, resp. are concurrent.
3. The parallels to NhaNoa, NhbNob, NhcNoc through Ha, Hb, Hc, resp. are concurrent.
4. The parallels to NhaNoa, NhbNob, NhcNoc through Na, Nb, Nc, resp. are concurrent.
5. The parallels to NhaNoa, NhbNob, NhcNoc through A', B', C', resp. are concurrent.

[Angel Montesdeoca]:

Correction to message Hyacinthos #25773

  1. The lines NhaNoa, NhbNob, NhcNoc are concurrent on the Euler line of HaHbHc
   

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

Midpoint of X(i) and X(j) for these {i,j}: {140,11808}, {143,1209}, {546,11802}, {6153,8254}.
With (6-9-13)-search numbers :{3.47121760910561,-1.41150180267242,3.01575760263193}

 2. The parallels to NhaNoa, NhbNob, NhcNoc through A, B, C, resp. are concurrent.
 

W2 = X(54) = (a^2(-b^2 c^2 (b^2-c^2)^2-(b^2-c^2)^2 (b^2+c^2)a^2+3 (b^4+b^2 c^2+c^4)a^4-3 (b^2+c^2) a^6+a^8) : ... : ... : ...)

   3. The parallels to NhaNoa, NhbNob, NhcNoc through Ha, Hb, Hc, resp. are concurrent.
    (corrected)

W3 = (a^2 (-(b^2-c^2)^4 (b^6+c^6)+2 (b^2-c^2)^2 (2 b^8-b^4 c^4+2 c^8) a^2-(b^2-c^2)^2 (5 b^6+7 b^4 c^2+7 b^2 c^4+5 c^6) a^4-2 b^2 c^2 (b^4+b^2 c^2+c^4) a^6+(5 b^6+8 b^4 c^2+8 b^2 c^4+5 c^6)a^8-2 (2 b^4+3 b^2 c^2+2 c^4)a^10+(b^2+c^2)a^12) : ... : ... )

Midpoint of X(i) and X(j) for these {i,j}:  {4,6242}, {3519,6243}.
With (6-9-13)-search numbers : {12.4818250322888, -13.3344547819560,  7.11136777758982}

4. The parallels to NhaNoa, NhbNob, NhcNoc through Na, Nb, Nc, resp. are concurrent.
 (corrected)

W4 = (a^2(-(b^2-c^2)^6 (b^2+c^2)+(b^2-c^2)^2 (4 b^8-3 b^6 c^2-6 b^4 c^4-3 b^2 c^6+4 c^8)a^2+(-5 b^10+6 b^8 c^2+5 b^6 c^4+5 b^4 c^6+6 b^2 c^8-5 c^10)a^4+(-3 b^6 c^2-4 b^4 c^4-3 b^2 c^6)a^6+(5 b^6+8 b^4 c^2+8 b^2 c^4+5 c^6) a^8-2 (2 b^4+3 b^2 c^2+2 c^4)a^10+(b^2+c^2)a^12) : ... : ...)

Midpoint of X(i) and X(j) for these {i,j}:  {52,2888}, {1209,6152}.
With (6-9-13)-search numbers : {9.75987887362814, -8.04113571956090, 4.70304511531356}

    5. The parallels to NhaNoa, NhbNob, NhcNoc through A', B', C', resp. are concurrent.
    
W5 = (a^2(-(b^2-c^2)^4 (b^6+c^6)
+(b^2-c^2)^2 (4 b^8-b^6 c^2-3 b^4 c^4-b^2 c^6+4 c^8)a^2
+(-5 b^10+6 b^8 c^2+2 b^6 c^4+2 b^4 c^6+6 b^2 c^8-5 c^10)a^4
+(-5 b^6 c^2-4 b^4 c^4-5 b^2 c^6)a^6
+(5 b^6+9 b^4 c^2+9 b^2 c^4+5 c^6)a^8
-2 (2 b^4+3 b^2 c^2+2 c^4)a^10
+(b^2+c^2)a^12) : ... : ...)

Midpoint of X(i) and X(j) for these {i,j}:  {195,12280}, {6242,6288}, {6243,12325}
With (6-9-13)-search numbers : {18.9668860056214, -15.7641647682448, 5.80036962655939}

Angel Montesdeoca

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