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X(4), X(15), X(16), X(39), X(512), X(2896), X(3413), X(3414), X(15412),

X(39665), X(39666) : isogonal conjugates of X(2480), X(2479)

isogonal conjugates of the vertices of the reflected first Brocard triangle, see Table 32

other points below

Geometric properties :

(This is based on a private message by Kadir Altintas, 2020-09-18).

Let PaPbPc be the circumcevian triangle of P and denote by Ka, Kb, Kc the Lemoine points of triangles BCPa, CAPb, ABPc respectively.

If Ra, Rb, Rc are the reflections of Ka, Kb, Kc in the respective sidelines of ABC, then ABC and RaRbRc are perspective if and only if P lies on K1158. See K243 for related properties and a generalization below.

K1158 is invariant under the Cundy-Parry like transformation (see CL037) described as follows. Denote by P* the image of P in the isoconjugation with pole X(2489) that swaps X(4) and X(512).Let Q be the intersection of lines X(512)P and X(4)P*. When P lies on K1158, Q also lies on K1158 and obviously on the perpendicular L(P) at P to the Brocard axis. Note that P and Q lie on a same rectangular circum-hyperbola H(P) of ABC.

When L(P) is the Lemoine axis (L), H(P) is the Jerabek hyperbola (J) and P, Q are the isogonal conjugates of X(2479), X(2480).

When L(P) is the line at infinity, H(P) is the Kiepert hyperbola and P, Q are X(3413), X(3414). Hence K1158 has two real asymptotes which meet at X(39) on the cubic. The third asymptote is perpendicular to the Brocard axis and meets K1158 again at X, the second intersection of the line through X(4), X(2896) with the rectangular circum-hyperbola passing through X(i) for i = 4, 32, 237, 263, 511, 512, 2211, 2698, etc. Note that X is the tangential of X(39) in K1158.

K1158 meets the circumcircle at six points on pK(X6, X14712). Their three remaining common points are X(2896) and two points on the line through X(2), X(32) and on the circum-hyperbola passing through X(i) for i = 6, 187, 249, 512, 524, 598, 843, etc.

Let Ah, Ak be the traces on BC of the orthic line and Lemoine axis. K1158 meets the sideline BC again at A' which is the homothetic of Ah under h(Ak, -2). B', C' are defined likewise and the Cundy-Parry like transforms of these three points are the remaining points of K1158 on the altitudes of ABC.

The isogonal transform of K1158 is K1159.

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Generalization

Let A0, B0, C0 be the projections of Ka, Kb, Kc on the respective sidelines of ABC.

At is the image of Ka under the homothety h(A0, t) and define Bt, Ct cyclically.

The locus of P such that ABC and AtBtCt are perspective is a circum-cubic K(t).

K(-1) = K1158.

K(0) = K243 and K(1) is the union of the symmedians of ABC. These are the only pKs.

Six cubics are psKs namely K(0), K(1) as above, K(-3) = psK(X14479, X11058, ?), K(t→∞) = psK(X25, X2, ?), and two complicated cubics with pseudo-pivots X(1113), X(1114).