HYACINTHOS 25476

[Antreas P. Hatzipolakis]:

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

Denote:

Na, Nb, Nc = the NPC centers of OBC, OCA, OAB, resp.
N1, N2, N3 = the NPC centers of O'B'C', O'C'A', O'A'B', resp, where O ' = the circumcenter of A'B'C' [ = N]

(ie N1, N2, N3 are the complements of Na,Nb,Nc, resp.)

Ma, Mb, Mc = the midpoints of NaN1, NbN2, NcN3, resp.

The NPC center of MaMbMc lies on the Euler line of ABC.

[Peter Moses]:

Hi Antreas,

 

6 a^10-17 a^8 b^2+10 a^6 b^4+12 a^4 b^6-16 a^2 b^8+5 b^10-17 a^8 c^2+26 a^6 b^2 c^2-13 a^4 b^4 c^2+19 a^2 b^6 c^2-15 b^8 c^2+10 a^6 c^4-13 a^4 b^2 c^4-6 a^2 b^4 c^4+10 b^6 c^4+12 a^4 c^6+19 a^2 b^2 c^6+10 b^4 c^6-16 a^2 c^8-15 b^2 c^8+5 c^10::
on line {2,3}.

{X(140),X(2072)}-harmonic conjugate of X(3530).

Searches {3.167029015366514323432237183 86,2.2891543899599132844779188 5185,0.59415958945835560304953 3350977}.

 

 

Best regards,

Peter Moses.