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Any pK with pivot P=u:v:w, isopivot K (Lemoine point) and pole W = P x K (barycentric product) meets the circumcircle at A, B, C (where the tangents are the symmedians) and three other points Q1, Q2, Q3 (not necessarily all real) where the tangents are concurrent at a point X.

The first barycentric coordinate of X is :

a^2 [2 u v w (–13 b^2 c^2 u + a^2 c^2 v + a^2 b^2 w) + a^4 v^2 w^2 – 3 u^2 (c^4 v^2 + b^4 w^2)]

This type of pK is the isogonal transform of a pK with pivot G and pole the isogonal conjugate of P.

The pK always contains P, K, P/K (cevian quotient) and the crossconjugate of K and P (the isoconjugate of P/K).

Locus property :

Let S be the P–Ceva conjugate of K i.e. the perspector of the cevian triangle of P and the tangential triangle.

The locus of M such that the anticevian triangle of M and the circumcevian triangle of S are perspective (at Q) is the pK above. The locus of Q is another pK with same pole and pivot the barycentric product of the S–Ceva conjugate of K and tgP (isotomic of isogonal of P).

A simple example : with P = X(2) hence S = X(3), the two pKs above are both isogonal cubics, namely K002 and K004 respectively.

Special pK(P x K, P)

Circular cubic

The only circular cubic is obtained with P = X(671) and the pole is W = X(111). This is K273.

The two isotropic tangents meet at the singular focus F which lies on the tangent at X(111). F is X(11258), the reflection of O in X(111).

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Equilateral cubic

The only pK60 is K375. It is obtained when P is the isotomic conjugate of the intersection of the lines X(3)X(523) and X(315)X(524). The three tangents at Q1, Q2, Q3 make 60° angles with one another.

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pK+

The cubic has three concurring asymptotes (at X) if and only if P lies on a circumcubic passing through K (the cubic decomposes into the union of the symmedians), X(141), X(193), X(2998), the infinite points of pK(X39, X6), the reflections of the cevians of X(76) in the corresponding cevians of X(6). This gives the cubics pK(X39, X141) and pK(X3053, X193).

The locus of X is a cubic passing through X(2), X(6), X(3167), the infinite points of pK(X3, X6), the cevians of X(6).

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Cubics passing through the pole of isoconjugation

Every cubic K(P) = pK(P x K, P) with P on the Kiepert hyperbola contains its pole W = P x K and also G and H. The third point on the Euler line is the P-Ceva conjugate of X6.

This is the case of the Thomson cubic K002 and several other cubics shown in the following table.

Remarks :

Every such cubic K(P) meets the line at infinity and the circumcircle again at the same points as two isogonal pKs with pivots tP and atP respectively.

The tangents at the three points on the circumcircle are concurrent and the three lines passing through these points envelope the parabola (P) with focus X(111) and directrix the line GK. (P) passes through X{1499, 1640, 3288, 5915, 9209}.

The dual conic of (P) is a rectangular hyperbola (H), homothetic to the Kiepert hyperbola, passing through X{2, 20, 69, 76, 99, 3413, 3414, 3663, 6337, 9772, 11057, 20880, 22339, 22340, 32476}. Its center is the midpoint of X(69), X(99), also the reflection of X(115) in X(141).

(H) meets the circumcircle at X(99) and three other points on K169 = pK(X6, X69) and, more generally, on infinitely many pKs with pole on K177 = pK(X32, X2), pivot on K141 = pK(X2, X76), isopivot on K174 = pK(X32, X22).

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Cubics with pivot on the circumcircle

All the pK(P x K, P) with P on the circumcircle contain two other imaginary points Q1, Q2 on the circumcircle and on the Lemoine axis (the trilinear polar of K). The tangents at Q1, Q2 meet on the line KP.

In this case, P/K is the third point of the cubic on the Lemoine axis and its isoconjugate (P/K)* lies on the isogonal transform of the Steiner inellipse. The pole lies on the circumconic with perspector X(32).

Here is a selection of these cubics where P/K is in red and (P/K)* is in blue.

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pK passing through a given point Q distinct of K

pK(K x P, P) contains the given point Q if and only if P lies on the circumconic (C) passing through Q and the cevapoint (or cevian product) of K and Q. All these cubics form a pencil of pKs tangent at A, B, C to the symmedians and passing through K, Q and the crossconjugate Q' of K and Q.

Examples :

• if Q = G, we find Q' = H and (C) is the Kiepert hyperbola as seen above.
• if Q = I, we find Q' = X(57) and (C) is the circumconic through I and G.
• if Q = O, we find Q' = X(1073) and (C) is the circumconic through O and G.
• if Q = X(9), we find Q' = X(282) and (C) is the circumconic through X(9) and G.

Obviously, the Thomson cubic contains all these points.

Here is a selection of these cubics.

Generalization

CL043 is a sub-class of a more general type of pK intersecting the circumcircle at three points where the tangents are concurrent.

If W = p:q:r and P=u:v:w are the pole and the pivot of the pK, we must have the following conditions :

for a given pivot P, W lies on a quintic Q(P) passing through :

• A, B, C which are nodes, the tangents being the cevian lines of X(32) and the sidelines of the anticevian triangle of P x K
• P x K (barycentric product)
• P^2 (barycentric square) and the vertices of its cevian triangle

for a given pole W, P lies on a quintic Q(W) passing through :

• A, B, C which are nodes
• the square roots of W
• W÷K (barycentric quotient)
• the common points of the circumcircle and the trilinear polar of W÷K (barycentric quotient)
• the common points of the circumcircle and the line passing through W÷K and the crossconjugate of K and W÷K
• the vertices of the cevian triangle of Z, isoconjugate of the crossconjugate of K and W÷K in the isoconjugation with fixed point W÷K

for a given pole W, the isopivot S lies on a quartic Q(S) which is the isoconjugate of Q(W).

Their equations are downloadable as a text file.

For example, with W = K, we obtain the circular quintic Q063 and its isogonal conjugate Q113.