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Let pK(Ω, P) be the pivotal cubic with pole Ω = p:q:r and pivot P = u:v:w.

The polar conics of A, B, C in this pK are centered at Oa, Ob, Oc. In general, the triangles ABC and OaObOc are not perspective unless Ω lies on the pivotal cubic with pole P^2 x ctP and pivot G/P where P^2 is the barycentric square of P, ctP is the complement of the isotomic conjugate of P (and also the center of the inconic with perspector P), G/P is the G–Ceva conjugate of P (and also the center of the circumconic with perspector P). Here x denotes the barycentric product.

The most interesting case is obtained when Ω = ctP = P x cP since the triangles ABC and OaObOc are triply perspective. Any cubic pK(ctP, P) is a member of the class CL059. For example, with P = H we have Ω = K giving the Orthocubic K006.

The coordinates of these points are :

Oa = u(u+v)(u+w) : v(u+w)^2 : w(u+v)^2,

Ob = u(v+w)^2 : v(v+u)(v+w) : w(v+u)^2,

Oc = u(w+v)^2 : v(w+u)^2 : w(w+u)(w+v).

The perspectors of ABC and OaObOc are :

P0 = u(v+w)^2 : v(w+u)^2 : w(u+v)^2 = cP x ctP,

P1 = u / (u+w) : v / (v+u) : w / (w+v),

P2 = u / (u+v) : v / (v+w) : w / (w+u),

to compare with P x tcP = u / (v+w) : v / (w+u) : w / (u+v).

***

Construction and properties of these points

Let Qa be the intersection of the line A cP and the parallel at tcP to BC. Qb and Qc are defined similarly.

Let Q1 be the common point of the lines A tQc, B tQa, C tQb and Q2 the common point of the lines A tQb, B tQc, C tQa.

P1 and P2 are the barycentric products Q1 x P and Q2 x P respectively.

It follows that Oa = A P0 /\ B P2 /\ C P1, Ob = A P1 /\ B P0 /\ C P2 and Oc = A P2 /\ B P1 /\ C P0.

Let Ea, Eb, Ec be the related points defined as follows :

Ea = B Oc /\ C Ob = u(v+w) : v(u+v) : w(u+w),

Eb = C Oa / A Oc = u(u+v) : v(u+w) : w(v+w),

Ec = A Ob /\ B Oa = u(u+w) : v(v+w) : w(u+v).

These three points are clearly related to ctP = P x cP = u(v+w) : v(w+u) : w(u+v) and they are the P x ctP–isoconjugates of Oa, Ob, Oc respectively. This same isoconjugation swaps P1 and P2, G and P x ctP, P0 and P x tcP.

Note that the triangles OaObOc and EaEbEc are perspective at a point Q with barycentric coordinates

u(u^2+v^2+w^2+uv+uw+3vw) : v(u^2+v^2+w^2+vu+vw+3uw) :w(u^2+v^2+w^2+wu+wv+3uv),

showing that Q also lies on the line GP and on the line passing through ctP x cP and P x tcP. We shall meet this point again in the sequel.

Furthermore, the two following sets of points lie on a same conic :

Oa, Ob, Oc, P, P1, P2, ctP/P and Ea, Eb, Ec, P1, P2, ctP, P x ctP.

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Cubics related to triangles perspective to OaObOc

Let Y be the G–Hirst conjugate of cP i.e. the intersection of the line through G, P, cP with the polar line of cP in the Steiner circum-ellipse.

We have : Y = -u(u+v+w)+v^2+w^2+vw : : .

Theorem 1 : OaObOc is perspective with the cevian triangle of any point on the cubic pKc

with pole : cP x ctP^2 x Y,

with pivot : ctP ÷Y (barycentric quotient),

with isopivot : cP x ctP.

Theorem 2 : OaObOc is perspective with the anticevian triangle of any point on the cubic pKa

with pole : P x ctP,

with pivot : P x Y,

with isopivot : ctP ÷ Y.

Theorem 3 : in both cases, the locus of the perspector is the cubic pKp

with pole : P x ctP,

with pivot : Q defined above,

with isopivot : P x ctP ÷ Q.

Theorem 4 : pKc and pKa generate a pencil of cubics which contains a third pivotal cubic pK3

with pole : ctP^2 x Y,

with pivot : cP x ctP,

with isopivot : P x Y.

The following table gives a selection of points that lie on these cubics. pK is pK(ctP, P).

Recall that x and ÷ denote the barycentric product and quotient and that / denotes the cevian quotient.

ccP is the complement of the complement of P and Z is the G–Hirst conjugate of P.

Remarks

• pKc, pKa, pK3 have already seven known common points namely A, B, C, ctP, cP x ctP, P x Y, ctP÷Y (yellow points in the table). The remaining two common points are imaginary. They are the intersections of the trilinear polar of ctP and the circumconic with perspector ctP.

• the third points on the trilinear polar of ctP are the pink points in the table.

• the sixth points on the circumconic with perspector ctP are the blue points in the table.