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K1257

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X(114), X(371), X(372), X(511), X(13414), X(13415), X(46627), X(46628), X(46629), X(46630)

vertices Ha, Hb, Hc of the orthic triangle

foci of the orthic inconic

other points below

Geometric properties :

K1257 is a member of the pencil of cubics that contains K019, K048, K417, K418 and the cubic which is the union of the line at infinity and the axes of the orthic inconic.

K1257 is a focal cubic with singular focus X(114). Its orthic line is the Brocard axis and its real asymptote is the homothetic of the Brocard axis under h(X114, 2).

K1257 is an isogonal nK with respect to the orthic triangle hence it meets its sidelines at three collinear points.

If M* denotes the isogonal conjugate of M in this triangle, then K1257 is the locus of M such that the midpoint of MM* lies on the Brocard axis.

K1257 is also the locus of contacts of tangents drawn through X(114) to the circles passing through X(371), X(372).

K1257 meets the parallel at X(114) to the Brocard axis at Y, also on the line X(32), X(512) which is the perpendicular bisector of X(371), X(372).

K1257 meets its real asymptote at X = Y*, the tangential of X(114), on the line X(1976), X(2396).

K1257 is invariant under the JS transformation defined in Table 62. Hence, it is also a nK in the improper triangle X(511), X(13414), X(13415). The last points on its sidelines are Y13414 = X13414* and Y13415 = X13415*. These points are collinear with X(114) and lie on the lines {1114,13415}, {1113,13415} respectively. These lines are parallel to the Brocard axis.

These points Y, Y13414, Y13415, X are now X(46627), X(46628), X(46629), X(46630) in ETC (2022-01-12).