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X(115), X(1648), X(6784), X(39022), X(39023), X(66184), X(66185), X(66186), X(66187), X(66188), X(66189), X(66190), X(66191), X(66192), X(66260), X(66261), X(66262), X(66263), X(66264), X(66265), X(66266), X(66352)

more generally, barycentric products of X(115) and points on K1364

X(66186), X(66187) = Y3413, Y3414 are the products of X(3413), X(3414) and the tangentials of X(39022), X(39023)

infinite points of the sidelines of ABC

Geometric properties :

K1381 is the locus of the tripolar centroids of points on the circum-parabola with perspector X(115), called the X-parabola in ETC, see below. See the general case in CL045.

K1381 is also the barycentric product of X(115) and the cuspidal cubic K1364.

K1381 has three real asymptotes parallel to the sidelines of ABC. They are their images under the homothety h(X115, 1/3).

K1381 meets these sidelines again at six points on a same (blue dotted) parabola homothetic to the X-parabola under h(Z,3/4) where Z = (b^2-c^2)^2 (3 a^4-3 a^2 b^2-3 a^2 c^2+2 b^4-b^2 c^2+2 c^4) : : , SEARCH = 2.70960216445566, on the lines {50,230}, {115,523}, now X(66353) in ETC.

K1381 is a cuspidal cubic with cusp X(115) and cuspidal tangent perpendicular to the Euler line.

If two points on the X-parabola are collinear with X(690) i.e. lie on a parallel to the line {115,125}, then their tripolar centroids lie on a line passing through X(1648) which is a point of inflexion on K1381. Note that the tangents at X(1648) to K219 and K1381 coincide.

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The X-parabola (see ETC X12064)

• passes through : 476, 523, 685, 850, 892, 2395, 2501, 4024, 4036, 4581, 4608, 5466, 8599, 10412, 12065, 12079, 13636, 13722, 14775, 15328, 18808, 20578, 20579, 30508, 30509, 31065, 34246, 39240, 39241, 44768, 50946, 53153, 53154, 55199, 55201, 55253, 56321, 58784, 60029, 60042, 60043, 60055, 62519, 62631, 62632, 62645, 62672, 64217, 64258, 64935, 65539, 65559, 65716, 66267 → 66300.

• is the isogonal conjugate of the trilinear polar of X(249), the tangent to the circumcircle at X(110), through :

110, 351, 526, 684, 1576, 1624, 1634, 2421, 4556, 4558, 4636, 5467, 5502, 6132, 9138, 9145, 9216, 15329, 17402, 17403, 23181, 30510, 34291, 35327, 35329, 35330, 35357, 36829, 41880, 41881, 42741, 42742, 42743, 42744, 42745, 42746, 42747, 46616, 47053, 48953, 50947, 52603, 52605, 52606, 53263, 53280, 53295, 53301, 53306, 53309, 53315, 53322, 53324, 53326, 53384, 53385, 56980, 57119.

• is the isotomic conjugate of the trilinear polar of X(4590), the tangent to the Steiner circumellipse at X(99), through :

99, 110, 690, 2396, 3268, 3573, 4226, 4427, 4563, 4576, 4610, 5027, 5118, 5468, 5652, 6333, 8723, 9123, 9146, 9147, 9168, 10330, 10411, 14185, 14187, 15342, 17136, 17941, 24286, 30508, 30509, 30580, 32121, 34245, 35278, 35314, 35315, 35356, 39904, 45687, 48951, 48960, 52935, 53331, 53332, 53333, 53334, 53335, 53735, 55198, 55200, 55252, 56760, 57060, 57216, 57249.

• (more generally) is the P-isoconjugate of the tangent T(P) at Q = P x X(99) = P ÷ X(523) to the circum-conic C(P) with perspector P. This tangent is the trilinear polar of Q x X(99).

• is the dual conic of the inellipse with center X(620) and perspector X(4590) = barycentric square of X(99) = trilinear pole of the line {99, 110}. This ellipse is the barycentric square of the line {2,6} and passes through 2, 32, 439, 593, 1509, 2482, etc.