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X(3), X(4), X(6), X(468), X(2574), X(2575)

vertices Ha, Hb, Hc of the orthic triangle

traces A', B', C' of the orthic axis

points Q1, Q2, Q3 of (O) on pK(X6, X6515) and other cubics below

Every psK(P x H/P, H, P) is a nodal cubic with node P passing through the vertices Ha, Hb, Hc of the orthic triangle. Its pseudo-pole is the barycentric product of P and the H-Ceva conjugate of P. Its pseudo-pivot is H. Compare with the cubics psK(P x G/P, G, P) of CL033.

The nodal tangents at P are perpendicular if and only if P lies on Q187. This is the case of K429 = psK(X1974, X4, X6) obtained when P = X(6).

Q187 is circular with isotropic asymptotes concurring at X(468) on the curve. The two remaining asymptotes are real, parallel to the asymptotes of the Jerabek hyperbola at X = X(60774) on the lines {4,51}, {125,511}.

The tangents at A, B, C, H concur at O and the tangents at Ha, Hb, Hc concur at H.

Q187 meets (O) at A, B, C and three points Q1, Q2, Q3 which lie on the rectangular hyperbola (H) passing through X(4), X(6), X(25), X(110), X(1899), X(2574), X(2575) and X(6515) which is the orthocenter of the triangle Q1Q2Q3.

The inconic of ABC with perspector X(2052) is also an inconic of this triangle.

***

These points Q1, Q2, Q3 also lie on K163, K445, K621 and more generally on every cubic of a family of pKs with

• pole Ω on K1357 = pK(X36417 = X25^2, X393 = X4^2), passing through {6, 25, 32, 393, 1609, 2207, 44077},

• pivot S on K621 = pK(X393, X2052 = X92^2), passing through {2, 4, 6, 24, 393, 847, 2052, 6515},

• isopivot Q on K1358 = pK(X32 = X6^2, X24) passing through {3, 6, 24, 25, 1609, 2351, 9937, 34428}, which is a pK of the family.

When Ω = X(6), we find S = X(6515) and Q = gX(6515) = X(60775), on the lines {X3, X2165}, {X6, X1147}, {X24, X254}, {X25, X571}, {X37, X921}, {X159, X1976}.

When S = X(847), we find Ω = X(60778) on the lines {X5,X6}, {X25, X59189}, {X32, X14593} and Q = X(60776) on the lines {X3, X49}, {X6, X2351}, {X25, X571}, {X50, X161}, {X154, X3135}.

S1 = {X2,X311} /\ {X6,X847}, SEARCH = 2.00804107755034.

***

These points Q1, Q2, Q3 also lie on a set of nK0s with

• pole Ω on the trilinear polar of X(393) passing through {460, 512, 2501, 2520, 5140, 44084, 44705, 50387, 54273, 58552, 58757, 58895, 59991, 60428},

• root R on the trilinear polar of X(2052) passing through {403, 523, 6530, 14312, 14618, 15451, 17994, 23290, 37778, 39534, 39663, 44145, 44426, 46008, 51385, 52661, 57070, 58757, 59745, 59830, 59843, 59844, 59845, 59846, 59847, 59848, 59915, 59932, 59935}.

Example : nK0(X512, X403) passes through {6, 523, 2574, 2575, 8505, 8506}. See K624 for additional properties and other related cubics.

A generalization

Let Q be a point with cevian triangle QaQbQc and P another point not lying on the cevian lines of Q.

K(P, Q) = psK(P x Q/P, Q, P) is a nodal circum-cubic with node P passing through Qa, Qb, Qc. Recall that Q/P is the Q-Ceva conjugate of P, perspector of QaQbQc and the anticevian triangle of P.

The nodal tangents at P are perpendicular if and only if P lies on a circular quartic Q(P, Q) analogous to Q187.

When Q = G, Q(P, Q) decomposes into the line at infinity and the Thomson cubic K002. See the related CL033.

When Q ≠ G, Q(P, Q) passes through Q, Qa, Qb, Qc, the traces A', B', C' of the trilinear polar L(Q) of Q, the infinite points of the circum-conic C(Q) passing through Q and its isotomic conjugate tQ.

The tangents at A, B, C to K(P, Q) intersect at gQ, isogonal conjugate of Q.

The tangents at Qa, Qb, Qc concur if and only if Q is on K007. The point of concurrence is on K004.

Q(P, Q) meets the circumcircle again at three points Q1, Q2, Q3 which lie on pK(X6 x Q^2, Q) i.e. on the barycentric product Q x K002.

These points Q1, Q2, Q3 also lie on every cubic of a family of pKs with

• pole Ω on Ω(Q) = psK(X32 x Q x gcQ, Q x gcQ, X6 x Q^2),

• pivot S on S(Q) = psK(Q gcQ, Q tcQ, Q),

• isopivot T on a circum-cubic T(Q) passing through X6 x Q, the vertices of the tangential triangle KaKbKc with concurring tangents. The point of concurrence lies on the cubic if and anly if Q lies on K002, in which case the tangents at A, B, C to K(Q) also concur.

K(Q) is in fact the barycentric product of X(6) and the anticomplement of psK(gQ x cQ, X(2), cQ).

Remark 1 : when Q lies on K002, these three cubics are pKs. Ω(Q) passes through X(32), S(Q) and T(Q) pass through X(6).

Remark 2 : when Q lies on K007, Ω(Q), S(Q), T(Q) pass through X(6), X(4), X(3) respectively.