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K660

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X(7), X(320), X(527), X(903), X(3254)

Consider a point P with cevian triangle A1B1C1. Reflect A1 successively in the internal bisectors at C, A, B to obtain A2 and define B2, C2 likewise.

The triangles ABC and A2B2C2 are perspective (at Q) if and only if P lies on K660 and, in this case, Q also lies on K660 (César Lozada, ADGEOM #1087).

The line PQ is tangent at T to the parabola with focus X(150) and directrix the line X(1)X(7). The harmonic conjugate R of T with respect to P and Q is another point on K660.

The six points A1, B1, C1, A2, B2, C2 lie on the bicevian conic C(P,Q) whose center lies on the rectangular hyperbola passing through X(1), X(7), X(65), X(145), X(224), X(1071), X(1317), X(1537), X(3174), the vertices of the intouch triangle and whose asymptotes are parallel to those of the Feuerbach hyperbola. This hyperbola is actually the Jerabek hyperbola of the intouch triangle.

K660 is a nodal cubic with node X(7) and nodal tangents parallel to the asymptotes of the Feuerbach hyperbola. It meets the line at infinity at X(527) and two other imaginary points J1, J2 which are also on the circum-ellipse (C) with perspector the incenter X(1) and center X(9). The tangents at these points meet at X(3254) on the cubic. The real asymptote is the parallel at X(100) to the line X(2)X(7). In many ways, K660 is very similar to a strophoid.

Construction of K660 :

A variable line L through X(1156) – the reflection of X(7) about X(11) – meets (C) again at N and the line X(7)X(9) at D. The reflection M of N in the midpoint of X(1156)D is transformed into P under the translation which maps X(1156) onto X(7) and then P lies on K660.

If L is the tangent at X(1156) to (C) meeting the line X(7)X(9) at Y, then the translation maps Y onto X which is the intersection of K660 with its asymptote.