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X(3), X(30), X(52), X(1493), X(1992), X(2574), X(2575), X(3146), X(53777), X(53778), X(53779), X(53780), X(53781)

other points below

Geometric properties :

Let P be a point and let M be the orthocenter of its pedal triangle. When P traverses the Jerabek hyperbola (J), this point M lies on K1323.

Note that M is the barycentric product of the complement and the orthocorrespondent of a point on the Euler line.

K1323 is a nodal cubic with node X, on the lines {X3,X125}, {X4,X542}, {X52,X3627}, {X110,X578}, {X113,X137}, SEARCH = -3.09391068334600.

X is the common orthocenter of the two points on (J) which lie on the line (L) passing through X(23), X(6000), X(15054).

K1323 has three real asymptotes :

• one is parallel at X(125) to the Euler line. It is an inflexional asymptote since X(30) is the only real point of inflexion on the cubic.

• two are parallel to the asymptotes of the Jerabek hyperbola and meet at Y on the lines {X6,X110}, {X51,X1353}, {X52,X3627}, {X68,X265}, {X125,X343}, SEARCH = 1.01851125080067.

K1323 is invariant under the oblique symmetry with direction the Euler line and axis the (blue) line passing through X, Y, X(52), X(3627) on the Euler line, X(12295) on the first asymptote.

It follows that the tangent at X(52) to K1323 is parallel to the Euler line. More generally, any parallel to the Euler line meets K1323 at two points whose midpoint lies on the axis of the symmetry.

Other points on K1323

a^2 (a^4 b^2-b^6+a^4 c^2-2 a^2 b^2 c^2+b^4 c^2+b^2 c^4-c^6) (a^6-a^4 b^2-a^2 b^4+b^6-a^4 c^2-a^2 b^2 c^2-2 b^4 c^2-a^2 c^4-2 b^2 c^4+c^6) : : , SEARCH = 4.91989797754903

(2 a^6-5 a^4 b^2-2 a^2 b^4+5 b^6-5 a^4 c^2+12 a^2 b^2 c^2-5 b^4 c^2-2 a^2 c^4-5 b^2 c^4+5 c^6) (13 a^8-15 a^6 b^2-11 a^4 b^4+15 a^2 b^6-2 b^8-15 a^6 c^2+16 a^4 b^2 c^2-17 a^2 b^4 c^2-11 a^4 c^4-17 a^2 b^2 c^4+4 b^4 c^4+15 a^2 c^6-2 c^8) : : , SEARCH = 5.95647343574953

(3 a^4-2 a^2 b^2-b^4-2 a^2 c^2+2 b^2 c^2-c^4) (13 a^6-11 a^4 b^2-17 a^2 b^4+15 b^6-11 a^4 c^2+42 a^2 b^2 c^2-15 b^4 c^2-17 a^2 c^4-15 b^2 c^4+15 c^6) : : , SEARCH = 29.7366449260053

(4 a^4-5 a^2 b^2+b^4-5 a^2 c^2-2 b^2 c^2+c^4) (5 a^6-8 a^4 b^2+a^2 b^4+2 b^6-8 a^4 c^2-2 b^4 c^2+a^2 c^4-2 b^2 c^4+2 c^6) : : , SEARCH = 4.27801953376993

(a^4+a^2 b^2-2 b^4+a^2 c^2+4 b^2 c^2-2 c^4) (17 a^6-29 a^4 b^2+7 a^2 b^4+5 b^6-29 a^4 c^2+18 a^2 b^2 c^2-5 b^4 c^2+7 a^2 c^4-5 b^2 c^4+5 c^6) : : , SEARCH = 11.1769750783634

(2 a^4+a^2 b^2-3 b^4+a^2 c^2+6 b^2 c^2-3 c^4) (5 a^6-8 a^4 b^2+a^2 b^4+2 b^6-8 a^4 c^2+7 a^2 b^2 c^2-2 b^4 c^2+a^2 c^4-2 b^2 c^4+2 c^6) : : , SEARCH = 323.798598142681

a^2 (a^2-b^2-c^2)^2 (a^4-b^4+4 b^2 c^2-c^4) (a^2 b^2+b^4+a^2 c^2-2 b^2 c^2+c^4) : : , SEARCH = 4.82475388229153

(a^2-b^2-c^2) (5 a^2-b^2-c^2) (2 a^4-a^2 b^2-3 b^4-a^2 c^2+6 b^2 c^2-3 c^4) : : , SEARCH = -1.63483566041887

a^2 (2 a^2-b^2-c^2) (a^4-b^4+4 b^2 c^2-c^4) : : , SEARCH = -0.987656250922999