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 with 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.
Cubics related with 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.
• 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.