HYACINTHOS 27781

[Alexandr Skutin]:

Let ABC be a triangle.

Denote:

Na, Nb, Nc = the NPC centers of IBC, ICA, IAB, resp.

N1, N2, N3 = the isogonal conjugates of I wrt triangles NaBC, NbCA,
NcAB, resp.

1. N1, N2,N3 are collinear.

Which is this line ?

2. ABC, N1N2N3 are circumcyclologic.
ie the circumcircles of AN2N3, BN3N1, CN1N2 and ABC are concurrent
the circumcircles of N1BC, N2CA,N3AB and [degenerated] N1N2N3 [ = line
N1N2N3] are concurrent.

Cyclologic centers ?

 

[Angel Montesdeoca]:

**** 1. N1, N2,N3 are collinear.
Which is this line ?

a^6 (-b+c)+(b-c)^3 (b+c)^2 (b^2+b c+c^2)+a^4 (3 b^3-b^2 c+b c^2-3
c^3)+a^3 (-b^3 c+b c^3)+a b c (b^4+b^3 c-b c^3-c^4)-3 a^2 (b^5-c^5) x +
.... = 0.

{5, 79, 1749, 3336, 3467, 3652, 11246}

**** 2. ABC, N1N2N3 are circumcyclologic.

The cyclologic center (on circumcircle of ABC) of ABC with respect to
N1N2N3 is:

U = a^2/((b-c)(a^8 (b-c)^2+a^9 (b+c)+(b-c)^4 (b+c)^6+a (b-c)^2 (b+c)^5
(b^2+b c+c^2)-2 a^6 (b-c)^2 (2 b^2+3 b c+2 c^2)-a^7 (4 b^3+5 b^2 c+5 b
c^2+4 c^3)-a^2 (b^2-c^2)^2 (4 b^4+6 b^3 c+5 b^2 c^2+6 b c^3+4 c^4)+a^4
(b-c)^2 (6 b^4+16 b^3 c+21 b^2 c^2+16 b c^3+6 c^4)+a^5 (6 b^5+11 b^4 c+6
b^3 c^2+6 b^2 c^3+11 b c^4+6 c^5)-a^3 (4 b^7+11 b^6 c+6 b^5 c^2-6 b^4
c^3-6 b^3 c^4+6 b^2 c^5+11 b c^6+4 c^7))) : .... : ...

(6 - 9 - 13) - search numbers of U: (0.0573063005303804,
-0.0831477543240013, 3.67177925004082).

The cyclologic center of N1N2N3 with respect to ABC is:

V = a^13+2 a^12 (b+c)-(b-c)^6 (b+c)^7+a^9 b c (-5 b^2+4 b c-5
c^2)+a^11 (-3 b^2+2 b c-3 c^2)-a (b-c)^4 (b+c)^6 (2 b^2-b c+2 c^2)-a^10
(9 b^3+5 b^2 c+5 b c^2+9 c^3)+a^2 (b-c)^4 (b+c)^3 (3 b^4+4 b^3 c+3 b^2
c^2+4 b c^3+3 c^4)+5 a^8 (3 b^5+b^4 c+2 b^3 c^2+2 b^2 c^3+b c^4+3
c^5)+a^3 (b^2-c^2)^2 (9 b^6+8 b^5 c+2 b^4 c^2+b^3 c^3+2 b^2 c^4+8 b
c^5+9 c^6)+a^7 (10 b^6+6 b^5 c-3 b^4 c^2+10 b^3 c^3-3 b^2 c^4+6 b c^5+10
c^6)+2 a^4 b c (b^7+b^6 c-2 b^5 c^2-2 b^2 c^5+b c^6+c^7)-a^6 (10 b^7+4
b^6 c+3 b^5 c^2+b^4 c^3+b^3 c^4+3 b^2 c^5+4 b c^6+10 c^7)+a^5 (-15 b^8-8
b^7 c+12 b^6 c^2+b^5 c^3-7 b^4 c^4+b^3 c^5+12 b^2 c^6-8 b c^7-15 c^8) :
.... : ...

V lies on lines X(i)X(j) for these {i, j}: {5,79}, {30,5685}, {265,14452}.

(6 - 9 - 13) - search numbers of V: (0.820540785269206,
-0.513508968663262, 3.61745955931931).

Angel Montesdeoca