HYACINTHOS 24515

[Antreas P. Hatzipolakis]:

 

 

Let A be a triangle.

 

Denote:

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

Denote:

 

Nab, Nac = the orthogonal projections of Na on BNb, CNc, resp.

Nbc, Nba = the orthogonal projections of Nb on CNc, ANa, resp.

Nca, Ncb = the orthogonal projections of Nc on ANa, BNb, resp.

The circumcircles of NaNabNac, NbNbcNba, NcNcaNcb are concurrent.
The point of concurrence is the point of concurrence of ANa, BNb, CNc

Let Oa, Ob, Oc be the circumcenters of NaNabNac, NbNbcNba, NcNcaNc, resp.

ABC, OaObOc are orthologic.

[Peter Moses]:

Hi Antreas,

>The circumcircles of NaNabNac, NbNbcNba, NcNcaNcb are concurrent.

>The point of concurrence is the point of concurrence of ANa, BNb, CNc.

At X(1487).

>ABC, OaObOc are orthologic.

At X(1263)
OaObOc,ABC are othologic at:

14 a^16-103 a^14 b^2+335 a^12 b^4-633 a^10 b^6+765 a^8 b^8-609 a^6 b^10+313 a^4 b^12-95 a^2 b^14+13 b^16-103 a^14 c^2+454 a^12 b^2 c^2-729 a^10 b^4 c^2+342 a^8 b^6 c^2+459 a^6 b^8 c^2-786 a^4 b^10 c^2+469 a^2 b^12 c^2-106 b^14 c^2+335 a^12 c^4-729 a^10 b^2 c^4+360 a^8 b^4 c^4+15 a^6 b^6 c^4+480 a^4 b^8 c^4-837 a^2 b^10 c^4+376 b^12 c^4-633 a^10 c^6+342 a^8 b^2 c^6+15 a^6 b^4 c^6-14 a^4 b^6 c^6+463 a^2 b^8 c^6-758 b^10 c^6+765 a^8 c^8+459 a^6 b^2 c^8+480 a^4 b^4 c^8+463 a^2 b^6 c^8+950 b^8 c^8-609 a^6 c^10-786 a^4 b^2 c^10-837 a^2 b^4 c^10-758 b^6 c^10+313 a^4 c^12+469 a^2 b^2 c^12+376 b^4 c^12-95 a^2 c^14-106 b^2 c^14+13 c^16::

{6,9,13} Searches {1.782080963518729, 1.443932829199819, 1.818519770837398}.

on line {140, 930}.
Midpoint of X(140), X(1487).

 

Best regards,
Peter Moses.