X(1), X(3), X(4), X(376), X(3426), X(5119), X(7284), X(35237)
K243 is the isogonal pK with pivot the reflection X(376) of G in O. Hence, it is a member of the Euler pencil of cubics. See Table 27.
• The pedal triangle of P and the triangle of reflections of P in the vertices of ABC are perspective if and only if P lies on K243. See Table 6.
• Let PaPbPc be the circumcevian triangle of P and denote by Ka, Kb, Kc the Lemoine points of triangles BCPa, CAPb, ABPc respectively. Let Qa, Qb, Qc be the orthogonal proojections of Ka, Kb, Kc on BC, CA, AB respectively. ABC and QaQbQc are perspective (at Q) if and only if P lies on K243 (Kadir Altintas, private message, 2020-09-17). The locus of Q is a complicated isotomic circum-sextic passing through G (quadruple point with tangents passing through X(7) and its extraversions) and A, B, C are three nodes on the curve.
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.
If Ka, Kb, Kc are orthocenters instead of Lemoine points, the locus of P such that ABC and QaQbQc are perspective (at Q) is the Darboux cubic K004 and the locus of Q is another analogous (but simpler) isotomic circum-sextic passing through X(i) for i = 2, 4, 69, 459, 1440, 6616, 7080, 37669.