Conic through 40 ETC centers - Locus

The circumconic U with trilinear equation:

U : ∑ [ (b^2 – c^2)* v * w ] = 0

having center X(244) and perspector X(661) passes through A, B, C and these 40 ETC centers: X(1), X(10), X(19), X(37), X(65), X(75), X(82), X(91), X(158), X(225), X(267), X(596), X(759), X(775), X(876), X(897), X(921), X(969), X(994), X(1247), X(1581), X(1910), X(2153), X(2154), X(2166), X(2168), X(2186), X(2190), X(2214), X(2216), X(2217), X(2218), X(2219), X(2363), X(2588), X(2589), X(2652), X(2962), X(3668), X(4674).

U is the locus of P (not on the circumcircle of ABC) such that the antipedal triangle of P w/r to a ABC and the incentral triangle of ABC are orthologic.

X(759) is the fourth intersection of U and the circumcircle.

If O1(P) is the orthologic center (Incentral ; P-Antipedal ) for P then ( P, O1(P) )=( X(I), X(J) ) for these (I,J): (1,1), (10, 4065), (19,4319), (37, 2667), (65,2292), (75,192), (596,3159), (759,5497)

Some non-ETC O1(P) are:

P	O1(P)	Details of O1(P)
X(82)	(a^2+b^2+c^2-a*(b+c)+b*c)*(a^2+b^2-b*c+c^2)	On lines: (1,2896), (37,82), (192,3938), (744,2667), 3057,3100), (3159,3685), (4085,5262), (4514,4972)
X(876)	(a^2-b*c)*(b^2+c^2-a*(b+c))*(b-c)	Reflection of X(I) on X(J) for these (I,J): (2254,665), (3766,3716) On lines: (1,514), (37,513), (190,5378), (192,522), (512,2292), (523,2667), (649,4414), (659,4435), (665,1642), (764,4983), (891,3251), (1281,2785), (1742,3667), (3057,4083), (3716,3766), (4065,4151), (4124,4448)
X(897)	(a^2+b^2+c^2-3*b*c)*(a^2+b^2+c^2-3*a*(b+c)+3*b*c)	On lines: (1,2796), (37,100), (2836,3057), (4442,4956)
X(969)	(a^2+b^2+c^2+4*b*c)*(a^2+b^2+c^2+2*a*(b+c))	On lines: (1,193), (37,63), (192,612), (975,3159), (1773,3743), (2667,3870), (3920,4319), (4461,5297), (4657,5241)
X(994)	(a-2*(b+c))*(b^2+c^2+a*(b+c)-b*c)	On lines: (1,89), (2,4674), (8,3159), (37,517), (42,3899), (45,4752), (145,4065), (190,996), (192,519), (386,3878), (514,1000), (750,3245), (758,2667), (982,3898), (984,2802), (986,3884), (991,2800), (995,3877), (2099,4653), (2177,4867), (3670,3890), (3679,4125), (3727,3730), (3938,5497), (4256,5289)
X(2214)	(a^2+(b+c)^2)*(a^2+b^2+c^2+a*(b+c))	On lines: (1,69), (31,37), (192,3920), (344,1961), (534,4319), (612,2345), (1486,1962), (2667,3938), (3057,4336), (3159,3923), (3966,4657)

The locus of O1 is the conic V1 with trilinear equation:

V1: ∑ [ a*(b-c)*(u^2 + v*w) ] = 0

The center of V1 is:

   (b+c)*(a*(b+c)*(a^2-b*c)+(b^2-4*b*c+c^2)*a^2+2*b^2*c^2) : :

and lies on lines: (1,4427), (42,3952), (244,1962), (659,3722), (740,899), (1193,4065), (2292,2611), (2667,4117), (2802,3743)

V1 passes through the vertex of the incentral triangle and these 10 ETC centers: X(1), X(37), X(192), X(2292), X(2667), X(3057), X(3159), X(4065), X(4319), X(5497) .

Some non-ETC P (given O1(P)):

O1(P)	P	Details of P
X(37)	1/(a^2-a*(b+c)-b*c)	Isogonal conjugate of X(1621)  Midpoint of (3555,3696)  On lines: (10,141), (37,38), (65,1418), (75,3873), (81,82), (225,1876), (244,872), (596,740), (674,3664), (692,3449), (876,4132), (1002,4000), (1037,5228), (1468,2218), 1486,3423), (3286,3941), (3446,5091), (3555,3696), (3668,5173), 3681,4751), (3742,4698), (3779,4675), (4032,5083), (4430,4699)
X(3057)	1/(a^3-(b^2-b*c+c^2)*a-b*c*(b+c))	Isogonal conjugate of X(2975) On lines: (1,859), (5,10), (28,2190), (31,2217), (37,1953), (48,2214), (60,1610), (65,1193), (72,4692), (75,3869), (143,952), (197,1036), (214,5482), (225,1829), (595,759), (596,758), (957,3086), (961,3450), (1460,3435), (2218,3915), (2390,4292), (2933,5264), (3216,4674), (3668,3827)

The orthologic centers O2(P) (P-Antipedal ; Incentral) have complicated expressions. The only ETC-defined correspondence is O2( X(1) ) = X(3). The locus of O2(P) is also a conic with center of few interest.