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X(2), X(3), X(4), X(69), X(254), X(264), X(1993), X(5392)

isotomic conjugate of X(254)

vertices of antimedial triangle

vertices of the cevian triangle of X(264)

infinite points of pK(X6, X1993)

points of pK(X6, X11442) on the circumcircle

K045 is the isotomic pK with pivot X(264) = isotomic conjugate of X(3).

Locus properties :

1. locus of perspectors as seen in Euler central cubic.
2. locus of point P such that the cevian triangle of P and the orthic triangle are orthologic (Hyacinthos #8243-48). See a generalization at Table 7.
3. locus of point P such that the cevian triangle of P and the tangential triangle are orthologic.
4. locus of point P such that P, GSC(P), X(1993) are collinear. (X(1993) is the orthocorrespondent of X(3)). GSC is defined here.
5. Let A'B'C' be the cevian triangle of P. Ab, Ac are the pedals of A' on AC, AB and Ha is the orthocenter of AAbAc. Hb, Hc are defined similarly. Then the locus of P for which ABC and HaHbHc are orthologic is K045 (Paul Yiu, Hyacinthos #9863).
6. locus of pseudo-pivots of the stelloids psK60+ with asymptotes parallel to those of the McCay cubic K003. See K350.

The isogonal transform of K045 is K176. K045 is also related to Q033.

The complement of K045 is K612 = pK(X216, X2).

See other locus properties in the page K646.

***

A remarkable pencil :

K028 and K045 generate a pencil of cubics which are all psK(Ω, X264, X3) with pseudo-pole Ω on the Euler line.

The base-points are A, B, C, the vertices A', B', C' of the cevian triangle of X(264), X(3) and X(4) counted twice. Indeed, the tangent at X(4) is the same for every cubic (except K028 which is a nodal cubic with node H) and this tangent passes through X(52).

Every cubic meets

• the line at infinity at the same points as an isogonal pK with pivot P1 on the line {3, 54, 97, etc}.

• the circumcircle at the same points as an isogonal pK with pivot P2 on the line {4, 52, 58, etc}.

Special cubics of the pencil :

• K045, the only pK, namely pK(X2, X264).

• K028, the only stelloid and the only spK, with radial center X(381) and node X(4).

• K562, the only central cubic, with center X(5).

• K563, the only circular cubic, with singular focus X(14674).

• psK(X140, X264, X3), the only other K+ (apart K562) with asymptotes concurring at X on the line {5,51}.

Furthermore, this pencil contains three decomposed cubics. One of them is the union of the sideline BC (passing through A') and the conic passing through A, B', C', X(3) and X(4) with tangent through X(52). The two other cubics are defined likewise.