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X(2), X(23), X(94), X(111), X(323), X(3266), X(14919), X(18019), X(21907), X(32849), X(37798), X(46106)

X(46783) → X(46789)

X(52496) → X(52505)

X(52512) → X(52516)

points on (O) and on the de Longchamps line

Geometric properties :

K1303 is the isogonal transform of K381 = cK(#X6, X523). Its complement is K1438.

Locus properties

1. Let M be a point on the Jerabek hyperbola. The parallels at M to the cevian lines of X(110) meet the corresponding sidelines of ABC at three collinear points on the line (L). When M varies, the trilinear pole T of (L) lies on K1303.

The line MT envelopes a conic (C) with center X(125), bitangent to the Jerabek hyperbola at X(32618), X(32619) and tritangent to K1303. See figure 1.

2. Let (L1), (L2) be two lines passing through X(23) and symmetric in the Euler line. Let (C1), (C2) be their respective isotomic transforms. (L1), (C2) meet at M1, N1 and (L2), (C1) meet at M2, N2. These four points lie on K1303.

{M1, M2} and {N1, N2} are two pairs of isotomic conjugates. The vertices of the diagonal triangle of {M1, M2, N1, N2} are X(23), X(18019) and S on the line {2, 2485}. See figure 2.

Generalization of locus property 1

Let Q ≠ G be a point with complement cQ and isotomic conjugate tQ. Denote by C(Q) the circum-conic with center cQ, perspector ctQ.

Let M be a point on C(Q). The parallels at M to the cevian lines of Q meet the corresponding sidelines of ABC at three collinear points on the line (L). When M varies, the trilinear pole T of (L) lies on the cubic K(Q) = cK(#G, tQ).

(1) when Q lies on (O), C(Q) is a rectangular hyperbola and tQ lies on the de Longchamps axis.

(2) when Q lies on the Steiner ellipse, K(Q) is a cK0 and tQ, ctQ lie on (L∞). These are the cubics of CL031.

(3) when Q lies on the circum-conic with perspector X(599), the nodal tangents at G to K(Q) are perpendicular.

These three conditions are simultaneously realized if and only if Q = X(99). In this case, C(Q) is the Kiepert hyperbola and K(Q) is K185.

(4) KQ is a circular cubic if and only if tQ = X(11054) and Q =X(34898). The singular focus is X(14653).

(5) KQ is an equilateral cubic if and only if tQ = X(11055) and Q is unlisted. The asymptotes are parallel to those of K024.

(6) KQ has concurring asymptotes if and only if Q lies on the Tucker cubic T(-20).

K1304 = cK0(#X2, X522) and K1305 = cK0(#X2, X513) are two other examples.