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X(4), X(133), X(403), X(1986), X(5140), X(5146), X(5523), X(6000), X(10214), X(20410), X(44057), X(71346), X(71347), X(71348), X(71349), X(71350), X(71351), X(71352), X(71353), X(71354), X(71355), X(71356), X(71357), X(71358), X(71359), X(71360), X(71361), X(71362), X(71363), X(71370)

vertices of the orthic triangle

Geometric properties :

(contributed by Peter Moses)

The inverse in the circumcircle of the Jerabek hyperbola is K039.

Its inverse in the polar circle is the strophoid K1436 with node H and nodal tangents parallel to the asymptotes of the Jerabek hyperbola.

Compare K1436 and K1300 which is the inverse in the polar circle of the Kiepert hyperbola.

The singular focus of K1436 is X(133) on the nine point circle. Its polar conic is the circle passing through {4, 5, 133, 1539}.

The tangential X of X(133) is the inverse in the polar circle of the isogonal conjugate X31510* of X(31510). X lies on these lines : {4,9033}, {122,403}, {133,32743}, {31510,47087}.

This point X31510* lies on the lines {6,1636}, {64,526}, {74,520}, on the Jerabek hyperbola, on the circle through {3, 4, 64, 1301, 2693}.

The real asymptote of K1436 passes through X and X(6000) on the line at infinity. X(6000) lies on the lines {3,64}, {4,51}, {5,2883}, {6,1597}, {20,2979}, {24,1204}, {40,2939}, {52,382}, {65,1844} and many others.

X and X31510* are now X(71370) and X(71371) in ETC.

***

K1436 is also :

• the pedal curve with respect to H of the parabola (P) with focus X(107), directrix (D) passing through {4, 51, 185, 389, 1075, 1093, 1896, 1899, 2052, 3168, etc}.

• the isogonal cK with root X(512) with respect to the orthic triangle. Indeed, it meets the sidelines of the orthic triangle again at three collinear points on the line passing through X(526), X(1112).

• the barycentric product of H and the complement K1438 of K1303 = cK(#X2, X850).

• the locus of foci of conics inscribed in the orthic triangle whose center lies on the line X(4), X(51).

See the analogous strophoids K591, K955, K1300. See also CL077 for a generalization.