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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 circum-cubic K(P) passing through H and P*. The locus of the perspector is also a focal circum-cubic K'(P), passing through O and P, which is the isogonal transform of K(P).

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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 circum-hyperbola 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 F,P1 meets H(P ) again at P2 which lies on K(P).

• The orthic line L(P) of K(P) is the parallel at X(5) to the line OP*.

• The real asymptote A(P) of K(P) is parallel to L(P) hence K(P) passes through the infinite point ∞P of OP*.

A(P) is the image of L(P) under the homothety h(F, 2).

• The parallel at H to A(P) meets K(P) again at P3 on the lines F,P* and S,P1.

• The line OP* above meets K(P) again at P4 on the line H,F.

• The parallel at P to A(P) meets K(P) at P2 and another point P5 on the line S,P4.

• The line H,P1 meets H(P ) again at P6 on the line P*,P5.

• P7 = P1,P4 /\ P3,P5, P8 = ∞P,P1 /\ F,P5, P9 = H,P5 /\ P*,P1 are other points on K(P).

Remark : the midpoints the following pairs all lie on L(P) : {X4, P*}, {P1, P5}, {P2, P8}, {P3, P4}, {P6, P9}.

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Special cases

• When P lies on the line at infinity (L∞), K(P) must split into (L∞) and the rectangular circum-hyperbola 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, 38945, 38946, 38947, 38948, 38949, 38950, 38951, 38952, 39158, 39159, 39160, 39161, 39985, 39989, 39990, 39991, 39992, 39993, 41521, 42809, 42810, 47103, 47104, 47105, 47106, 47107, 47108, 47109, 47110, 47111, 52173, 52444, 52445, 52446, 52447, 61439, 61440, 61441, 61442, 61489}.

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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 circum-conic with perspector X(216) passing through X(110), X(265), X(1625). In this case, the pole lies on the circum-conic 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, K1280 = 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.

• K(P) is a central cubic if and only if P lies on a bicircular septic which is the antigonal transform of K026. In this case, the center, which is also the singular focus, is the antigonal of P. See K465 and K530.

• K(P) and K'(P) are strophoids if and only if P lies on Q106, passing through X(i) for i in {3, 4, 143, 265, 953, 24772}. See K025, K039 and K275, K274.

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" = inverse of P, 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. The singular focus of K"(P) is P.

These cubics form three pencils of focal cubics with base-points :

• K(P) : A, B, C, X(4), X(186) and circular points at infinity which are double since the tangents at these points concur at X(4).

• K'(P) : A, B, C, X(3), X(186) and circular points at infinity which are double since the tangents at these points concur at X(186).

• K"(P) : A, B, C, X(3), X(4), circular points at infinity and imaginary foci of the MacBeath inconic. These K"(P) are spK(P*, X5) as in CL055, and nK(X6, R, X3) with R on the trilinear polar of X(264). If S is the second point of the line HP on (O), then R is the H-isoconjugate of S, equivalently the barycentric quotient S* ÷ O.

This trilinear polar of X(264) passes through {297, 525, 850, 2501, 2592, 2593, 3569, 3580, 4391, 5523, 9979, 10015, 13302, 14316, 14618, 14918, 17434, 17773, 17896, 17924, 18314, 21438, 24978, 25259, 26545, 26546, 33294, 39905, 41079, 44146, 44427, 45266, 46106, 46107, 46108, 46109, 46110, 47286, 48380, 48381, 50188, 51358, 51481, 52744, 54074, 54262, 54395, 57043, 57065, 57224, 57257}.

See the cubics nK(X6, R, X3) in CL062 and also Table 80. K"(P) is globally invariant under both Cundy-Parry transformations Phi and Psi, see CL037. The lines OP and HP meet K"(P) again at P1 = Phi(P) and P2 = Psi(P) which lie on K028 and K009 respectively. P1 and P2 are isogonal conjugates on K"(P).

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Additional cubics (computed by Peter Moses)