[Antreas P. Hatzipolakis]:
Let ABC be a triangle and A'B'C' the orthic triangle.
The reflections of OA in HA, HB, HC bound a triangle Ta
The reflections of OB in HA, HB, HC bound a triangle Tb
The reflections of OC in HA, HB, HC bound a triangle Tc
Denote:
La, Lb, Lc = the Euler lines of Ta,Tb,Tc,resp.
Na, Nb, Nc = the NPC centers of Ta,Tb,Tc, resp.
1.La,Lb, Lc are concurrent.
2. the reflections of La,Lb,Lc in BC, CA, AB concur (at the line of infinity)
3. the parallels of La, Lb, Lc through A',B',C', resp. are concurrent
4. The circumcenter of NaNbNc lies on the Euler line of ABC.
[Peter Moses]:
Hi Antreas,
1).
a^22-5 a^20 b^2+10 a^18 b^4-13 a^16 b^6+22 a^14 b^8-42 a^12 b^10+56 a^10 b^12-50 a^8 b^14+33 a^6 b^16-17 a^4 b^18+6 a^2 b^20-b^22-5 a^20 c^2+18 a^18 b^2 c^2-21 a^16 b^4 c^2+2 a^14 b^6 c^2+22 a^12 b^8 c^2-41 a^10 b^10 c^2+57 a^8 b^12 c^2-64 a^6 b^14 c^2+53 a^4 b^16 c^2-27 a^2 b^18 c^2+6 b^20 c^2+10 a^18 c^4-21 a^16 b^2 c^4+9 a^14 b^4 c^4+11 a^12 b^6 c^4-17 a^10 b^8 c^4-a^8 b^10 c^4+35 a^6 b^12 c^4-55 a^4 b^14 c^4+43 a^2 b^16 c^4-14 b^18 c^4-13 a^16 c^6+2 a^14 b^2 c^6+11 a^12 b^4 c^6+4 a^10 b^6 c^6-6 a^8 b^8 c^6-12 a^6 b^10 c^6+17 a^4 b^12 c^6-18 a^2 b^14 c^6+15 b^16 c^6+22 a^14 c^8+22 a^12 b^2 c^8-17 a^10 b^4 c^8-6 a^8 b^6 c^8+16 a^6 b^8 c^8+2 a^4 b^10 c^8-33 a^2 b^12 c^8-6 b^14 c^8-42 a^12 c^10-41 a^10 b^2 c^10-a^8 b^4 c^10-12 a^6 b^6 c^10+2 a^4 b^8 c^10+58 a^2 b^10 c^10+56 a^10 c^12+57 a^8 b^2 c^12+35 a^6 b^4 c^12+17 a^4 b^6 c^12-33 a^2 b^8 c^12-50 a^8 c^14-64 a^6 b^2 c^14-55 a^4 b^4 c^14-18 a^2 b^6 c^14-6 b^8 c^14+33 a^6 c^16+53 a^4 b^2 c^16+43 a^2 b^4 c^16+15 b^6 c^16-17 a^4 c^18-27 a^2 b^2 c^18-14 b^4 c^18+6 a^2 c^20+6 b^2 c^20-c^22::
on lines {{128, 10539}, ...}.
2).
X(1510).
3).
X(137).
4).
X(10224).
Best regards,
Peter Moses.
The reflections of OA in HA, HB, HC bound a triangle Ta
The reflections of OB in HA, HB, HC bound a triangle Tb
The reflections of OC in HA, HB, HC bound a triangle Tc
Denote:
La, Lb, Lc = the Euler lines of Ta,Tb,Tc,resp.
Na, Nb, Nc = the NPC centers of Ta,Tb,Tc, resp.
1.La,Lb, Lc are concurrent.
2. the reflections of La,Lb,Lc in BC, CA, AB concur (at the line of infinity)
3. the parallels of La, Lb, Lc through A',B',C', resp. are concurrent
4. The circumcenter of NaNbNc lies on the Euler line of ABC.
[Peter Moses]:
Hi Antreas,
1).
a^22-5 a^20 b^2+10 a^18 b^4-13 a^16 b^6+22 a^14 b^8-42 a^12 b^10+56 a^10 b^12-50 a^8 b^14+33 a^6 b^16-17 a^4 b^18+6 a^2 b^20-b^22-5 a^20 c^2+18 a^18 b^2 c^2-21 a^16 b^4 c^2+2 a^14 b^6 c^2+22 a^12 b^8 c^2-41 a^10 b^10 c^2+57 a^8 b^12 c^2-64 a^6 b^14 c^2+53 a^4 b^16 c^2-27 a^2 b^18 c^2+6 b^20 c^2+10 a^18 c^4-21 a^16 b^2 c^4+9 a^14 b^4 c^4+11 a^12 b^6 c^4-17 a^10 b^8 c^4-a^8 b^10 c^4+35 a^6 b^12 c^4-55 a^4 b^14 c^4+43 a^2 b^16 c^4-14 b^18 c^4-13 a^16 c^6+2 a^14 b^2 c^6+11 a^12 b^4 c^6+4 a^10 b^6 c^6-6 a^8 b^8 c^6-12 a^6 b^10 c^6+17 a^4 b^12 c^6-18 a^2 b^14 c^6+15 b^16 c^6+22 a^14 c^8+22 a^12 b^2 c^8-17 a^10 b^4 c^8-6 a^8 b^6 c^8+16 a^6 b^8 c^8+2 a^4 b^10 c^8-33 a^2 b^12 c^8-6 b^14 c^8-42 a^12 c^10-41 a^10 b^2 c^10-a^8 b^4 c^10-12 a^6 b^6 c^10+2 a^4 b^8 c^10+58 a^2 b^10 c^10+56 a^10 c^12+57 a^8 b^2 c^12+35 a^6 b^4 c^12+17 a^4 b^6 c^12-33 a^2 b^8 c^12-50 a^8 c^14-64 a^6 b^2 c^14-55 a^4 b^4 c^14-18 a^2 b^6 c^14-6 b^8 c^14+33 a^6 c^16+53 a^4 b^2 c^16+43 a^2 b^4 c^16+15 b^6 c^16-17 a^4 c^18-27 a^2 b^2 c^18-14 b^4 c^18+6 a^2 c^20+6 b^2 c^20-c^22::
on lines {{128, 10539}, ...}.
2).
X(1510).
3).
X(137).
4).
X(10224).
Best regards,
Peter Moses.
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