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Circular pK(W, H)

Let P be the pivot of a pK. When P is different of H, there is one and only one circular such pK but when P = H, there are infinitely many such cubics.

These cubics have their singular focus F on the nine point circle and their pole W on the orthic axis. The isoconjugate H* of H is the point at infinity of the cubic.

pK(W, H) is also the locus of M whose orthotransversal has a fixed direction, that of any perpendicular to the direction given by H*.

The intersection with the real asymptote is X, the antipode of F on the nine point circle and this asymptote envelopes the Steiner deltoid H3. The "last" point on the circumcircle is E orthoassociate of X.

These cubics are isogonal pivotal circular cubics with respect to the orthic triangle.

They are also invariant under orthoassociation i.e. inversion in the polar circle and under three other inversions with poles A, B, C swapping H and the feet Ha, Hb, Hc of the relative altitude. They are also antigonal cubics i.e. cubics invariant under antigonal conjugation. They are the isogonal transforms of the inversible cubics.

All circular pK(W, H) form a pencil of circular cubics which contains three focal cubics : the pole must be a common point Wa, Wb, Wc of the orthic axis and a symmedian of ABC. For example pK(Wa, H) has singular focus Ha, meets its real asymptote at Xa antipode of Ha on the nine point circle. This asymptote is parallel to OA.

See also the related quartics at Q107.

The following table gives a selection of circular pK(W, H) with corresponding X and E.

Remark : the centers Oa, Ob, Oc of the osculating circles at A, B, C to any pK(W, H) above lie on three perpendiculars to the real asymptote. In other words, the triangles ABC and OaObOc are perspective with perspector at infinity. OaObOc is also perspective to HaHbHc with perspector on a complicated sextic.

Other remarkable pK(W, H)

Let K = pK(W, H) be a pivotal cubic with isopivot H* not lying on the line at infinity. Recall that K is an isogonal pivotal cubic with pivot H* with respect to the orthic triangle. The isogonal transform of pK(W, H) is a pK with isopivot O, a member of CL074.

• K is a pK+ i.e. has concurring asymptotes if and only if W lies on K208. In this case, H* (blue cells in the table) lies on K071 and the asymptotes concur on K412.

• When the pole W (green cells in the table) lies on the Brocard axis of the orthic triangle T, H* lies on the Euler line of T and the cubic is a member of the Euler pencil of T. It contains X(5) and X(52). This Brocard axis passes through X(i) for i = 5, 53, 216, 233, 3199, 6116, 6117, 6750, 8887, 10003, 10600, 11062, 14576, 14577, 14640, 27358, 27359, 27371, 32428, 35719, 36412, 39530, 39569.

• pK(W, H) contains its pole W if and only if W lies on the Euler line and then H* lies on the line GK (pink cells in the table). These cubics belong to a same pencil and always contain G and X(193).

M-OAP points

These points are studied in Table 53. We simply recall that any pK with pivot H, isopivot H* (not lying on the line at infinity) contains the four M-OAP points where M is the reflection of H* about the circumcenter O.

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Additional cubics by Peter Moses

Bi-isogonal pK(W, H)

Recall that K= pK(W, H) is an isogonal pivotal cubic with pivot H* = W ÷ H with respect to the orthic triangle. The isopivot is the H-Ceva conjugate of H*. It is the tangential of H* in pK(W, H).

When W lies on K627 = pK(X3199, X393), this cubic is also an isogonal pK with respect to another triangle T and then the cubic is said to be a bi-isogonal pK with a second pivot Q.

See Bi-isogonal and Tri-isogonal Pivotal Cubics.

Points of pK(W, H) on (L∞) and (O)

Every cubic pK(W, H) meets the line at infinity (L∞) and the circumcircle (O) at the same points as two isogonal pKs with respective pivots P1, P2.

Obviously, when W = X(6), P1 = P2 = H corresponding to the Orthocubic K006.

When W ≠ X(6), P1 = a(Ω ÷ H) and P2 = at(cevapoint(Ω ÷ X6, H)). See CL073 for a generalization.

pK(W, H) and pK(X6, P1) meet at three points on (L∞) and six points on a circum-conic passing through X(2).

pK(W, H) and pK(X6, P2) meet at six points on (O) and three points on a line passing through X(6), the isogonal transform of the circum-conic above.

Let Q be the midpoint of P1, P2. Q is the X(25)-isoconjugate of the Ceva-point of X(6) and W. The cubic spK(P1, Q) meets (L∞) and (O) at the same points as pK(X6, P1) and pK(X6, P2) respectively, hence pK(W, H) and spK(P1, Q) share the same points on (L∞) and (O).

spK(P1, Q) is a pK if and only if W lies on the bicevian conic C(X4, X112), then P1 lies on the Jerabek hyperbola, P2 lies on the bicevian conic C(X4, X648), and X(6), P1, P2 are collinear. In such case, this pK is an isogonal cubic with respect to the orthic triangle and its pivot is H*.

Example : K1184 = pK(X1841, X4) = spK(X72, X44547) and P2 = X(14054), H* = X(942).

With W = X6, X25, X53, X427, X571, we obtain K006, K233, K049, K517, K1318 respectively.