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X(468), X(523), X(10006), X(10011), X(22264)

U, V, W : traces of the trilinear polar (L) of X(393)

K869 is a circular nodal cubic mentioned in ETC, preamble of X(10001).

Its singular focus F = X(14693) is the midpoint of X(5), X(187). The polar conic of F is a circle (C) passing through the node X(468).

K869 is acnodal (figure above), crunodal, cuspidal depending on whether ABC is acutangle, obtusangle, rectangle.

K869 is the locus of M, the midpoint of PQ, where P is a variable point on the orthic axis and Q its G-Ceva conjugate on the nine point circle. In other words, P is the perspector of a rectangular circum-hyperbola (H) and Q is its center. See a generalization below.

For example,

• P = X(523), Q = X(115), M = X(523) and (H) is the Kiepert hyperbola.

• P = X(650), Q = X(11), M = X(10006) and (H) is the Feuerbach hyperbola.

• P = X(647), Q = X(125), M = X(22264) and (H) is the Jerabek hyperbola.

• P = X(230), Q = X(114), M = X(10011).

Note that X(468) is obtained when P is a (real or not) common point of the circumcircle and the orthic axis.

The intersection X of K869 with its real asymptote is obtained when P = X(2492), Q = X(5099) and (H) passing through X4, X23, X316, X842, X1383, etc. X is unlisted with SEARCH = 1.29389728627773.

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Generalization

Let (L) be a line with tripole Z ≠ G. For P on (L), the G-Ceva conjugate Q of P lies on the bicevian conic C(G, Z). Q is the center (resp. perspector) of the circum-conic with perspector (resp. center ) P.

The midpointt M of P and Q lies on a singular cubic K(Z) with node N on (L) which is the barycentric product of Z and the infinite point of the line G, tZ.

K(Z) meets the line at infinity at the infinite points of (L) and the circum-conic C(Z) with perspector ctZ, the complement of the isotomic conjugate of Z. Note that C(Z) is a rectangular hyperbola when Z lies on (O).

K(Z) meets the sidelines of ABC at the traces of the trilinear polare of Z^2 and six points on an ellipse homothetic to the Steiner ellipse whose center lies on the line G, tZ^2.

K(Z) is acnodal, crunodal when Z lies inside, outside the Steiner ellipse. It is a cuspidal cubic with cusp G when Z lies on the Steiner ellipse i.e. when (L) passes through G. See K1364 for example obtained when Z = X(99) hence C(Z) is a rectangular hyperbola.