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K869

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

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.

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 is unlisted in ETC with SEARCH = –0.793292366869797 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.