HYACINTHOS 26366

[Le Viet An]:

Dear Mr Rodinos

Let ABC be a triangle and L the Euler line.

Denote:

E = the point of concurrence of the reflections of L in BC, CA, AB, resp. [ =X(110)]
P = the reflection of E in O [ = X(74)]

The lines passing through P and parallels to OA,OB,OC intersect L at Pa,Pb,Pc, resp.
The perpendiculars to APa, BPb, CPc, at Pa,Pb, Pc, resp. bound a triangle A'B'C'.

1. The NPC of A'B'C' passes through O.
Which is its center?
2.The orthocenter of A'B'C' lies on the circumcircle of ABC.
Which point is it?

Thank you very much.
Best regards,
Le Viet An.

[Peter Moses]:

Hi Antreas,

1) 

a^2 (a^26-4 a^24 b^2+21 a^20 b^6-23 a^18 b^8-45 a^16 b^10+120 a^14 b^12-78 a^12 b^14-45 a^10 b^16+98 a^8 b^18-56 a^6 b^20+9 a^4 b^22+3 a^2 b^24-b^26-4 a^24 c^2+28 a^22 b^2 c^2-47 a^20 b^4 c^2-68 a^18 b^6 c^2+291 a^16 b^8 c^2-219 a^14 b^10 c^2-273 a^12 b^12 c^2+585 a^10 b^14 c^2-357 a^8 b^16 c^2+19 a^6 b^18 c^2+68 a^4 b^20 c^2-25 a^2 b^22 c^2+2 b^24 c^2-47 a^20 b^2 c^4+235 a^18 b^4 c^4-260 a^16 b^6 c^4-479 a^14 b^8 c^4+1318 a^12 b^10 c^4-839 a^10 b^12 c^4-350 a^8 b^14 c^4+655 a^6 b^16 c^4-251 a^4 b^18 c^4+12 a^2 b^20 c^4+6 b^22 c^4+21 a^20 c^6-68 a^18 b^2 c^6-260 a^16 b^4 c^6+1162 a^14 b^6 c^6-967 a^12 b^8 c^6-1127 a^10 b^10 c^6+2196 a^8 b^12 c^6-964 a^6 b^14 c^6-128 a^4 b^16 c^6+149 a^2 b^18 c^6-14 b^20 c^6-23 a^18 c^8+291 a^16 b^2 c^8-479 a^14 b^4 c^8-967 a^12 b^6 c^8+2852 a^10 b^8 c^8-1587 a^8 b^10 c^8-943 a^6 b^12 c^8+1075 a^4 b^14 c^8-199 a^2 b^16 c^8-20 b^18 c^8-45 a^16 c^10-219 a^14 b^2 c^10+1318 a^12 b^4 c^10-1127 a^10 b^6 c^10-1587 a^8 b^8 c^10+2578 a^6 b^10 c^10-773 a^4 b^12 c^10-220 a^2 b^14 c^10+75 b^16 c^10+120 a^14 c^12-273 a^12 b^2 c^12-839 a^10 b^4 c^12+2196 a^8 b^6 c^12-943 a^6 b^8 c^12-773 a^4 b^10 c^12+560 a^2 b^12 c^12-48 b^14 c^12-78 a^12 c^14+585 a^10 b^2 c^14-350 a^8 b^4 c^14-964 a^6 b^6 c^14+1075 a^4 b^8 c^14-220 a^2 b^10 c^14-48 b^12 c^14-45 a^10 c^16-357 a^8 b^2 c^16+655 a^6 b^4 c^16-128 a^4 b^6 c^16-199 a^2 b^8 c^16+75 b^10 c^16+98 a^8 c^18+19 a^6 b^2 c^18-251 a^4 b^4 c^18+149 a^2 b^6 c^18-20 b^8 c^18-56 a^6 c^20+68 a^4 b^2 c^20+12 a^2 b^4 c^20-14 b^6 c^20+9 a^4 c^22-25 a^2 b^2 c^22+6 b^4 c^22+3 a^2 c^24+2 b^2 c^24-c^26):: 

on lines {{74, 186}, {520, 13293}}.

2). X(1304).

Best regards,
Peter Moses.