Interesting locus: Conic through 34 ETC centers

What is the locus of P such that the anticevian triangle of P w/r to ABC is perspective to the intangents triangle of ABC?

The answer is:

1) The antiorthic axis of ABC, line (X(44)X(513)), with trilinear equation u+v+w = 0, or

2) The circumhyperbola U with trilinear equation

U: ∑ [ a*(b-c)*(b+c-a)*v*w ] = 0

The perspector of U is X(663).

The center of U is:

a*(b-c)^2*(b+c-a)*(a^3-2*(b+c)*a^2+(b^2+b*c+c^2)*a-b*c*(b+c)) : :

and lies on lines (X(I)X(J)) for these (I,J): (41,4557), (649,4014), (657,3271), (1200,2246), (3124,5075)      

U pasess through A, B, C and these 34 ETC centers: X(6), X(9), X(19), X(55), X(57), X(284), X(333), X(673), X(893), X(909), X(1024), X(1174), X(1436), X(1751), X(1945), X(2160), X(2161), X(2164), (2195), X(2258), X(2259), X(2291), X(2299), X(2316), X(2319), X(2337), X(2339), X(2343), X(2364), X(2432), X(2590), X(2591), X(3451), X(3512) 

Mappings:

1) If P is on the antiorthic axis of ABC then the perspector Z(P) lies on line (X(650)X(663)) with trilinear equation: u/(s-a)+v/(s-b)+w/(s-c)=0. This line passes through this 15 ETC-centers: X(650), X(663), X(861), X(2340), X(3689), X(3900), X(4041), X(4105), X(4162), (4433), X(4435), X(4477), X(4814), X(4895), X(4959)

There are 92 known ETC centers on the antiorthic axis (up to X(5546)) and for only two of them the correspondant perspectors are defined in ETC. These [ P, Z(P) ] are [X(650), X(4162)] and [X(652), X(663)].

You can see/download the mapping from this part of the locus in table 1 and table 1.2 at end of this page, [Note: most of pairs (P, Z(P)) are not easy to handle and only the most simple ones has been calculated.]

2) If P lies on conic U, then the perspectors Z(P) lie on the hyperbola V with trilinear equation:

V: ∑ [ a*(b-c)*(b+c-a)^2*v*w ] = 0

The perspector of V is X(657).

The center of V has trilinears:

a*(b-c)^2*(-a+b+c)^2*(a^2*(a^2-b^2+b*c-c^2)-a*(b+c)*(-b^2-c^2+a^2)-b*c*(b-c)^2) : :

or, equivalently:

cos(A/2)^2*(cos(2*A)+r/R+3)*sin(B/2-C/2)^2 : :

and lies on lines (X(I)X(J)) for these (I,J): (1,4566), (11,3138), (663,3022), (997,4319), (1015,3269), (1064,2293), (1984,2310)

The hyperbola V passes through A, B, C and these 14 ETC centers: X(1), X(33), X(55), X(64), X(103), X(200), X(220),X(963), X(1043), X(2192),X(2328), X(2332), X(2342), X(4845).

For P on cubic U, (P, Z(P)) = (X(I),X(J)) for these [I,J]:   [6, 55], [9, 200], [19, 33], [55, 220], [57, 1], [284, 2328], [333, 1043], [909, 2342], [1436, 2192], [2291, 4845], [2299, 2332] .

Please see/download tables 2.1 and 2.2  for complete lists of points and perspectors.

  Direct mapping for P on antiorthic axis   Download

  Inverse mapping for Z(P) on line (X(650)X(663))   Download

  Direct mapping for P on conic U   Download

  Inverse mapping for Z(P) on conic V   Download

Note: Show tables seems to work well only with Google Chrome.