HYACINTHOS 28850

[Antreas P. Hatzipolakis]:

1. NPCs version:

Let ABC be a triangle,

Denote:

A', B', C' = the midpoints of AN, BN, CN, resp.

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

D = the Poncelet point of ABCN = X(137)

The circumcircles of DA'Na, DB'Nb, DC'Nc are coaxial.

2nd (other than D) intersection?


2. Circumcircles version (by taking A'B'C' as reference triangle):
:
Let ABC be a triangle.

Denote:

(O1) ,(O2), (O3) = the circumcircles of NBC, NCA, NAB, resp.

(Oa) ,(Ob), (Ob) = the reflections of  (O1) ,(O2), (O3) in BC, CA, AB, resp.
They concurr at D = antigonal conjugate of N = X(1263).

The circumcircles of DAOa, DBOb, DCOc are coaxial.

2nd (other than D) intersection?


[Angel Montesdeoca]:

*** 1. 2nd (other than D=X(137)) intersection is X(14051)
*** 2.  2nd (other than D=X(1263)) intersection is W =

= REFLECTION OF X(5) IN X(14051)

 = (a^4+(b^2-c^2)^2-a^2 (b^2+2 c^2)) (a^12-(b^2-c^2)^6-a^10 (4 b^2+3 c^2)+a^8 (5 b^4+5 b^2 c^2+3 c^4)-a^6 (3 b^4 c^2+2 b^2 c^4)+a^2 (b^2-c^2)^2 (4 b^6-6 b^4 c^2-3 b^2 c^4+3 c^6)+a^4 (-5 b^8+9 b^6 c^2+b^4 c^4+4 b^2 c^6-3 c^8))  :  : 

= lies on these lines: {4,93}, {5,930}, {17,8173}, {18,8172}, {30,252}, {54,1263}, {143,11538}, {195,20414}, {265,6798}, {550,1487}, {1157,28237}, {1879,2937}, {2070,21394}, {3459,6150}, {11671,15345}, {20030,20424}, {20413,27090}, {21230,24306}.
 
= reflection of X(i) in X(j), for these {i, j}: {5,14051}, {930,19268}.

 (6 - 9 - 13) - search numbers  of W: (-6.86065907646552, -6.08754885586978, 11.0215794174168).
 
 Angel Montesdeoca