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Let P, Q be two distinct points with cevian triangles PaPbPc, QaQbQc respectively. Denote by Oa, Ob, Oc the circumcenters of triangles APaQa, BPbQb, CPcQc. When P is the orthocenter H of ABC, these points Oa, Ob, Oc are clearly the midpoints of AQa, BQb, CQc hence the triangles ABC and OaObOc are perspective at Q for all Q. In the sequel, we suppose that P ≠ H is a fixed point and denote by K(P) the locus of Q such that ABC and OaObOc are perspective and by K'(P) the locus of the perspector. From the remark above, K(P) must contain H and K'(P) must contain P. When Q is the isogonal conjugate P* of P, the circles APaQa, BPbQb, CPcQc are tangent at A, B, C respectively to the circumcircle (O) of ABC hence ABC and OaObOc are perspective at O on K'(P) then P* must be on K(P). Main theorem For P ≠ H, the locus of Q such that the triangles ABC and OaObOc are perspective is a focal circumcubic K(P) passing through H and P*.The locus of the perspector is also a focal circumcubic K'(P), passing through O and P, which is the isogonal transform of K(P). *** Properties of K(P) • The singular focus F of K(P) is the antigonal of P i.e. the antipode of P in the rectangular circumhyperbola H(P) passing through P. The center of H(P) is denoted Ω. Hence, if the singular focus of K(P) is F ≠ H, then the singular focus of K(F) is P and each cubic is the antigonal transform of the other. Furthermore, the singular focus of K'(P) is the inverse of F in (O). • K(P) passes through P1, the midpoint of H and aP, where aP is the anticomplement of P. • The line passing through aΩ and P1 meets (O) at aΩ and another point S which lies on K(P). • The line FP1 meets H(P )again at P2 which lies on K(P). • The real asymptote A(P) of K(P) is parallel to the line OP* hence K(P) passes through the infinite point of OP*. • The parallel at H to A(P) meets K(P) again at P3 on the lines FP* and SP1. • The line OP* above meets K(P) again at P4 on the line HF. • The parallel at P to A(P) meets K(P) at P2 and another point P5 on the line SP4. • The line HP1 meets H(P )again at P6 on the line P*P5. • P7 = P1P4 /\ P3P5 is another point on K(P). *** Special cases • When P lies on the line at infinity (L∞), K(P) must split into (L∞) and the rectangular circumhyperbola which is the isogonal transform of the line OP. • When P lies on (O), the center Ω of H(P) lies on the nine point circle and then F = H. In this case, K(P) contains X(265) and the reflection of P in the Euler line (E) of ABC. The real asymptote passes through O. • K(X265) splits into (O) and (E). Hence, for any P ≠ H on (E), K(P) passes through X(265) and F lies on K025 which is K(X3) with singular focus X(265). • K(P) passes through P (and then K'(P) passes through P*) if and only if P lies on Q038, a circular quintic passing through X(i) for i in {1, 4, 5, 80, 1113, 1114, 1263, 2009, 2010}. • K(P) passes through O (and then K'(P) passes through H) if and only if P lies on K025, a strophoid passing through X(i) for i in {4, 30, 265, 316, 671, 1263, 1300, 5080, 5134, 5203, 5523, 5962, 10152, 11604, 11605, 11703, 13495, 16172, 19552, 31862, 31863, 34150, 34169, 34170, 34171, 34172, 34173, 34174, 34175, 34239, 34240, 37888}. *** Special types of K(P) • K(P)is a K0 (no term in x y z) if and only if P lies on K337 = pK(X232, X4) passing through X(i) for i in {4, 114, 371, 372, 511, 2009, 2010, 3563, 5000, 5001, 32618, 32619}. • K(P)is a nK if and only if P lies on the circumconic with perspector X(216) passing through X(110), X(265), X(1625). In this case, the pole lies on the circumconic with perspector X(51), the root lies on a complicated quartic passing through X(2), X(648), X(18883). The isogonal conjugate of the singular focus lies on the circle with diameter OH. For example, K(X110) = nK(X1989, X18883, X4) is the isogonal transform of K933. • K(P)is a psK if and only if P lies on a complicated tricircular nonic passing through H and the feet Ha, Hb, Hc of the altitudes. • In particular, K(Ha), K(Hb), K(Hc) are three focal pKs with singular foci Ha, Hb, Hc respectively. 



The following table gives a selection of these focal cubics K(P) and K'(P), with singular foci F and F', for P ≠H not lying on the line at infinity. K"(P), with singular focus F", is the inverse of K'(P) in the circumcircle and also the cubic K(F) where F = antigonal(P) is the singular focus of K(P). 



The lines in yellow are those with P on (O) hence the singular foci of K(P) and K'(P) are X(4) and X(186) respectively. *** Additional cubics (computed by Peter Moses) 




