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X(1), X(515), X(1317), X(1319), X(1323), X(11700), X(15524), X(15730), X(33901), X(33902), X(33903), X(37743), X(39751), X(39752), X(39753), X(39754), X(39755), X(39756), X(39757), X(39758), X(39759), X(39760), X(39761), X(39762), X(39763), X(56741)

X(63769) → X(63777)

vertices of the intouch triangle A'B'C'

Geometric properties :

K1160 is a strophoid with node X(1) and nodal tangents parallel to the asymptotes of the Feuerbach hyperbola. Its singular focus F is X(11700), the midpoint of X(1), X(109). The polar conic of X(11700) is the circle that also passes through X(1) and X(1385), the midpoint of X(1)X(3).

The real asymptote of K1160 is the image of the line X(1)X(4) under the homothety h(X109, 3/2). It meets the cubic at X = X(39763), also on the line X(1)X(1769).

K1160 is the inverse in the incircle of the rectangular hyperbola (H) with center X(5083), passing through X(i) for i = 1, 7, 65, 145, 224, 1071, 1317, 1537, 3174, 3307, 3308, etc, hence homothetic to the Feuerbach hyperbola. (H) is the Jerabek hyperbola of the intouch triangle, also the bicevian conic C(X7, X664) with respect to ABC.

K1160 is also the pedal curve of X(1) with respect to the inscribed parabola (P) with perspector X(664), focus X(109), directrix the line passing through X(1) and X(4). (P) passes through X(522), X(3676), X(4105) and is tangent to the nodal tangents above at two points on the line X(100)X(109). This line passes through the points of inflexion of K1160 but one only is real. (P) is the dual curve of the circum-hyperbola with perspector X(522), center X(1146), passing through X(i) for i = 2, 8, 29, 85, 92, 178, 189, 257, 312, 333, 1121, 1220, 1311, 1952, 2090, 2399, 2988, 2994, etc.

K1160 is the locus of contacts of tangents drawn through the singular focus X(11700) to the circles tangent at X(1) to the line X(1)X(4), the orthic line of K1160.

Locus property

Let P be a point and let Pa, Pb, Pc be its reflections in the sidelines BC, CA, AB respectively. The triangles A'B'C' and PaPbPc are perspective (at Q) if and only if P lies on K1160, and then, Q also lies on K1160. P and Q are conjugated on the cubic : the parallel at P to the asymptote meets K1160 again at R and the line FR meets K1160 again at Q. Here is a selection of such pairs {P,Q} computed by Peter Moses, see Euclid #6220 :

{1,1}, {515,11700}, {1317,1319}, {1319,1317}, {1323,15730}, {11700,515}, {15524,63770}, {15730,1323}, {33901,63771}, {33902,63772}, {33903,63773}, {37743,63774}, {39751,63769}, {39752,39753}, {39753,39752}, {39754,39755}, {39755,39754}, {39756,39758}, {39757,39759}, {39758,39756}, {39759,39757}, {39760,63775}, {39761,63776}, {39762,39763}, {39763,39762}, {63769,39751}, {56741,63777}.

Generalization

Now A'B'C' is the pedal triangle of a point X ≠ H. The locus property above also gives a strophoid S(X) with node X and nodal tangents parallel to the asymptotes of the rectangular circum-hyperbola passing through X.

All the properties of K1160 are easily adapted. For instance, the singulat focus F of S(X) is the midpoint of X and the isogonal conjugate of the infinite point of the perpendiculars to the line HX.

With X = X(3), we obtain the Stammler strophoid K038.

Remark : when X = H, the triangles A'B'C' and PaPbPc are perspective for every P in the plane and Q is the H-Ceva conjugate of P.