HYACINTHOS 25773

 

[Antreas P. Hatzipolakis]:

 

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]:

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) : ... : ...)

 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.
   
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) : ... : ... )

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

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) : ... : ...)

    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) : ... : ...)

Angel Montesdeoca