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See also K584 for a trilinear equation and related comments

X(2), X(6), X(110), X(523), X(5967), X(5968),

bicentric pair P(4), U(4) : intersections of the circumcircle and nine point circle (not always real) and their isogonal conjugates

Ha, Hb, Hc projections of G on the altitudes and their isogonal conjugates Oa, Ob, Oc. These points are the vertices of the orthocentroidal and orthocentroidal-isogonic triangles. They also lie on K005. See below for another equation related to K005

other points and a more detailed figure below

K397 is the locus of roots of equilateral nK0+ cubics. The locus of their poles is K396. The locus of the radial center is Q153.

K397 is also the locus of orthopivots R of all orthopivotal cubics O(R) which are nK cubics. See §6.2.2 in the FG paper "Orthocorrespondence and orthopivotal cubics" in the Downloads page and also Orthopivotal cubics.

K397 is the isogonal cubic nK(X6, X30, X2). It is a member of the class CL061.

K397 has three asymptotes : one is the line X(523)X(1989) and the two other are parallel to the asymptotes of the circumconic with center K, perspector O.

K397 contains :

X = X(14998), on the real asymptote.

X* = X(14999), its isogonal conjugate, the third point on GK.

Y = X(5967), intersection of the lines X(2)X(110) and X(6)X(523).

Y* = X(5968), intersection of the lines X(6)X(110) and X(2)X(523).

K397 is the cubic Ke1(X523) in CL054.

K396 and K397 generate a pencil of circum-cubics which also contains pK(X512, X6).

The following table gives a selection of these related cubics.

point R on K397

orthopivotal O(R)

equilateral nK0+(W, R)


Kiepert hyperbola + line at infinity

infinitely many with W at infinity, all decomposed into the line at infinity and a circum-conic


K018 Brocard (second) cubic

K024 = Kjp


circumcircle + Fermat line









Remark : K005 and K397 have 9 identified common points namely A, B, C, Ha, Hb, Hc, Oa, Ob, Oc which is confirmed by the similarity of their equations involving cyclic products..

K005 : (c^2 y - 2 SA z) - (b^2 z - 2 SA y) = 0

K397 : (c^2 y - 2 SA z) + (b^2 z - 2 SA y) = 0

Compare with K001 and K216 :

K001 : (c^2 y + 2 SA z) - (b^2 z + 2 SA y) = 0

K216 : (c^2 y + 2 SA z) + (b^2 z + 2 SA y) = 0