HYACINTHOS 24436

[Antreas P. Hatzipolakis]:

 

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

Denote:

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

M1, M2, M3 = the midpoints of IA', IB', IC', resp.

MaMbMc, M1M2M3 are cyclologic.

(ie the circumcircles of MaM2M3, MbM3M1, McM1M2 are concurrent and the circumcircles of M1MbMc, M2McMa, M3MaMb are concurrent)

Cyclologic centers?

 

[Peter Moses]:

Hi Antreas,

 

>the circumcircles of MaM2M3, MbM3M1, McM1M2 are concurrent at
2 a^9-2 a^8 b-3 a^7 b^2+a^6 b^3+a^5 b^4+5 a^4 b^5-a^3 b^6-5 a^2 b^7+a b^8+b^9-2 a^8 c+8 a^7 b c-a^6 b^2 c-2 a^5 b^3 c-3 a^4 b^4 c-12 a^3 b^5 c+9 a^2 b^6 c+6 a b^7 c-3 b^8 c-3 a^7 c^2-a^6 b c^2+2 a^5 b^2 c^2-2 a^4 b^3 c^2+a^3 b^4 c^2+19 a^2 b^5 c^2-16 a b^6 c^2+a^6 c^3-2 a^5 b c^3-2 a^4 b^2 c^3+24 a^3 b^3 c^3-23 a^2 b^4 c^3-6 a b^5 c^3+8 b^6 c^3+a^5 c^4-3 a^4 b c^4+a^3 b^2 c^4-23 a^2 b^3 c^4+30 a b^4 c^4-6 b^5 c^4+5 a^4 c^5-12 a^3 b c^5+19 a^2 b^2 c^5-6 a b^3 c^5-6 b^4 c^5-a^3 c^6+9 a^2 b c^6-16 a b^2 c^6+8 b^3 c^6-5 a^2 c^7+6 a b c^7+a c^8-3 b c^8+c^9::
{{1,1537},{55,108},{123,3816}, ...}.

 

>the circumcircles of M1MbMc, M2McMa, M3MaMb are concurrent at X(1387).

 

Best regards,

Peter Moses.