HYACINTHOS 24853

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle.

Denote:

Na, Nb, Nc =the NPC centers of NBC, NCA, NAB, resp.

Oa, Ob,Oc = the reflections of O in BC, CA, AB, resp.

NaNbNc, OaObOc are perspective.

[Peter Moses]:

Hi Antreas,

 

>NaNbNc, OaObOc are perspective.
a^16-3 a^14 b^2-a^12 b^4+14 a^10 b^6-20 a^8 b^8+9 a^6 b^10+3 a^4 b^12-4 a^2 b^14+b^16-3 a^14 c^2+16 a^10 b^4 c^2-10 a^8 b^6 c^2-18 a^6 b^8 c^2+17 a^4 b^10 c^2+a^2 b^12 c^2-3 b^14 c^2-a^12 c^4+16 a^10 b^2 c^4-3 a^8 b^4 c^4-18 a^6 b^6 c^4-13 a^4 b^8 c^4+21 a^2 b^10 c^4-2 b^12 c^4+14 a^10 c^6-10 a^8 b^2 c^6-18 a^6 b^4 c^6-14 a^4 b^6 c^6-18 a^2 b^8 c^6+19 b^10 c^6-20 a^8 c^8-18 a^6 b^2 c^8-13 a^4 b^4 c^8-18 a^2 b^6 c^8-30 b^8 c^8+9 a^6 c^10+17 a^4 b^2 c^10+21 a^2 b^4 c^10+19 b^6 c^10+3 a^4 c^12+a^2 b^2 c^12-2 b^4 c^12-4 a^2 c^14-3 b^2 c^14+c^16 on lines {{5,7691},{140,1141},...}.

 

Best regards,

Peter Moses.


[Angel Montesdeoca]:

Dear Antreas,

The perspector of triangles NaNbNc and  OaObOc is W = X(5)X(7691) /\ X(140)X(1141)

W = ( b^16-3 b^14 c^2-2 b^12 c^4+19 b^10 c^6-30 b^8 c^8+19 b^6 c^10-2 b^4 c^12-3 b^2 c^14+c^16+
(-4 b^14+b^12 c^2+21 b^10 c^4-18 b^8 c^6-18 b^6 c^8+21 b^4 c^10+b^2 c^12-4 c^14) a^2+
(3 b^12+17 b^10 c^2-13 b^8 c^4-14 b^6 c^6-13 b^4 c^8+17 b^2 c^10+3 c^12) a^4+
(9 b^10-18 b^8 c^2-18 b^6 c^4-18 b^4 c^6-18 b^2 c^8+9 c^10) a^6+
(-20 b^8-10 b^6 c^2-3 b^4 c^4-10 b^2 c^6-20 c^8) a^8+
(14 b^6+16 b^4 c^2+16 b^2 c^4+14 c^6) a^10+
(-b^4-c^4) a^12+
(-3 b^2-3 c^2) a^14+
a^16 : ... : ...),

with (6-9-13)-search numbers  (-9.15830277618339,  -1.00277730161421,  8.56180389510963).

Best regards,
Angel Montesdeoca

Angel Montesdeoca