HYACINTHOS 26038

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle and A'B'C' the pedal triangle of O.

Denote:

Ma, Mb, Mc = the midpoints of AA', BB', CC', resp.

Maa, Mab, Mac = the reflections of Ma in OA', OB', OC', resp.

Mba, Mbb, Mbc = the reflections of Mb in OA', OB', OC', resp.

Mca, Mcb, Mcc = the reflections of Mc in OA', OB', OC', resp.

(Na), (Nb), (Nc) =  NPCs of MaaMabMac, MbaMbbMbc, McaMcbMcc, resp.

1, (Na), (Nb), (Nc) are concurrent at a point on the Euler line of NaNbNc.

Which point is it wrt triangle: 

a. ABC 

b. NaNbNc ?

 

 

[Peter Moses]:

 

 

Hi Antreas,

 

1a.

a^2 (a^2-b^2-c^2) (a^6 b^2-a^4 b^4-a^2 b^6+b^8+a^6 c^2-2 a^4 b^2 c^2+2 a^2 b^4 c^2+b^6 c^2-a^4 c^4+2 a^2 b^2 c^4-4 b^4 c^4-a^2 c^6+b^2 c^6+c^8):: 

on lines {{2,1112},{3,74},{20,12133},{1 13,6823},{125,343},{140,9826}, {143,6640},{146,10996},{265,66 43},{339,4576},{376,12292},{39 4,13198},{511,5159},{542,10691 },{631,1986},...}.

complement X(1112).

midpoint of X(i) and X(j) for these {i,j}: {{3,12358},{20,12133},{974,556 2},{1216,6699},{2979,12099},{ 6101,12236},{12219,13148}}.

reflection of X(i) in X(j) for these {i,j}: {{9826, 140}, {11746, 6723}}.

 

1b.

X(2071).

 

Best regards,

Peter Moses.