HYACINTHOS 25358

[Antreas P. Hatzipolakis]:

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

Denote:

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

1. The circumcircles of OA'Ma, OB'Mb, OC'Mc are coaxial.

2. the crcumcircles of HA"Ma, HB"Mb, HC"Mc are coaxial.

(2nd point of intersection on the Euler line)

 

2nd points of intersections?

 

[Peter Moses]:

Hi Antreas,

 

1) a^2 (a^2-b^2-c^2) (a^6-3 a^4 b^2+3 a^2 b^4-b^6-3 a^4 c^2-2 a^2 b^2 c^2+b^4 c^2+3 a^2 c^4+b^2 c^4-c^6) (a^10 b^2-5 a^8 b^4+10 a^6 b^6-10 a^4 b^8+5 a^2 b^10-b^12+a^10 c^2+4 a^8 b^2 c^2-8 a^6 b^4 c^2+6 a^4 b^6 c^2-9 a^2 b^8 c^2+6 b^10 c^2-5 a^8 c^4-8 a^6 b^2 c^4+8 a^4 b^4 c^4+4 a^2 b^6 c^4-15 b^8 c^4+10 a^6 c^6+6 a^4 b^2 c^6+4 a^2 b^4 c^6+20 b^6 c^6-10 a^4 c^8-9 a^2 b^2 c^8-15 b^4 c^8+5 a^2 c^10+6 b^2 c^10-c^12)::
on line {{3,49},...}.

 

2) X(5159).

 

Best regards,    

Peter Moses