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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).

***

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.

***

pK+

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

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Cubics passing through the pole of isoconjugation

All the pK(P x K, P) with P on the Kiepert hyperbola contain the pole W = P x K and also G and H. This is the case of the Thomson cubic K002 and several other cubics shown in the following table.

pivot

centers on the cubic

cubic

X2

X1, X2, X3, X4, X6, X9, X57, X223, X282, X1073, X1249

K002

X4

X2, X4, X6, X25, X193, X371, X372, X2362

K233

X10

X2, X4, X6, X10, X42, X71, X199, X1654

 

X13

X2, X4, X6, X13, X15, X62, X1251, X2306, X3129, X3180

 

X14

X2, X4, X6, X14, X16, X61, X3130, X3181

 

X76

X2, X4, X6, X22, X69, X76, X1670, X1671

K141

X83

X2, X4, X6, X83, X251, X1176

K644

X94

X2, X4, X6, X94, X265, X1989, X2070

 

X98

X2, X4, X6, X98, X237, X248, X385, X1687, X1688, X1976

K380

X226

X2, X4, X6, X73, X226, X1400

 

X262

X2, X4, X6, X262, X263, X1689, X1690, X3148

K791

X275

X2, X4, X6, X54, X275, X1993

 

X321

X2, X4, X6, X37, X72, X321, X2895, X2915

 

X598

X2, X4, X6, X598, X1383, X1992, X1995

K283

X671

X2, X4, X6, X23, X111, X524, X671, X895

K273

X801

X2, X4, X6, X20, X394, X801

 

X1446

X2, X4, X6, X1427, X1439, X1446

 

X1916

X2, X4, X6, X39, X256, X291, X511, X694, X1432, X1916

K354

X2052

X2, X4, X6, X24, X393, X847, X2052

K621

X2394

X2, X4, X6, X2394, X2433

 

X2592

X2, X4, X6, X2574, X2592

 

X2593

X2, X4, X6, X2575, X2593

 

X2986

X2, X4, X6, X30, X323, X2986

 

X(14534)

X2, X4, X6, X21, X58, X81, X572, X961, X1169, X1220, X1798, X2298

K379

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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.

pivot

centers on the cubic

cubic

X74

X6, X15, X16, X74, X1495

 

X98

X2, X4, X6, X98, X237, X248, X385, X1687, X1688, X1976

K380

X100

X6, X100, X667, X1016

 

X101

X6, X101, X649, X1252

 

X103

X6, X55, X103, X672

 

X105

X1, X6, X57, X105, X238, X1438, X2195, X2223

 

X106

X6, X106, X902, X2226

 

X109

X6, X109, X663, X1262

 

X110

X6, X110, X249, X512

 

X699

X6, X32, X699, X1691

 

X727

X6, X31, X727, X1914, X3009

 

X733

X6, X83, X251, X694, X733

 

X741

X6, X58, X81, X292, X741, X1326, X1911, X2106

 

X1297

X3, X6, X511, X1073, X1297

 

X1298

X6, X54, X275, X1298, X1987

 

X1477

X6, X56, X1458, X1477

 

X2249

X6, X284, X1172, X1945, X1949, X2249

 

***

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.

pivot

centers on the cubic

cubic

cubics passing through X(9) and X(282)

X2

X1, X2, X3, X4, X6, X9, X57, X223, X282, X1073, X1249

K002

X9

X6, X9, X198, X259, X282, X2066

 

X200

X6, X9, X55, X200, X282

 

X281

X6, X9, X19, X281, X282, X2331

 

X346

X6, X9, X282, X346, X1604, X2324

 

X2287

X6, X9, X219, X282, X284, X610, X1172, X2287

 

X2297

X6, X9, X282, X1449, X2297

 

 

 

 

cubics passing through O and X(1073)

X2

X1, X2, X3, X4, X6, X9, X57, X223, X282, X1073, X1249

K002

X97

X3, X6, X54, X97, X275, X577, X1073

 

X394

X3, X6, X219, X222, X394, X1073, X1433, X1498

 

X1214

X3, X6, X65, X1073, X1214

 

X1297

X3, X6, X511, X1073, X1297

 

 

 

 

cubics passing through I and X(57)

X1

X1, X6, X55, X57, X365, X1419, X2067

 

X2

X1, X2, X3, X4, X6, X9, X57, X223, X282, X1073, X1249

K002

X28

X1, X6, X28, X57, X1474, X1724, X2299, X2352

 

X57

X1, X6, X56, X57, X266, X289, X1743

 

X81

X1, X6, X57, X58, X81, X222, X284, X1172, X1433

K319

X88

X1, X6, X36, X44, X57, X88, X106, X1168, X1465, X2226, X2316

K454

X89

X1, X6, X57, X89, X999, X2163, X2364

 

X105

X1, X6, X57, X105, X238, X1438, X2195, X2223

 

X274

X1, X6, X57, X86, X274, X333

 

X277

X1, X6, X57, X277, X2191

 

X278

X1, X6, X19, X34, X57, X278, X1723

 

X279

X1, X6, X57, X269, X279, X1617

 

X291

X1, X6, X42, X57, X239, X291, X292, X672, X894, X1757, X1967

K135

X330

X1, X6, X57, X87, X330, X2319

 

X959

X1, X6, X57, X959, X1460, X2258

 

X961

X1, X6, X57, X961, X1402

 

X985

X1, X6, X31, X57, X985, X2280

 

X1002

X1, X6, X57, X1002, X2279

 

X1170

X1, X6, X57, X218, X1170, X1174

 

X1219

X1, X6, X57, X1219, X2297

 

X1255

X1, X6, X35, X37, X57, X1126, X1171, X1255

 

X1257

X1, X6, X57, X72, X1257, X2983

 

X1258

X1, X6, X57, X171, X213, X1258

 

X1280

X1, X6, X57, X518, X1280

 

X1390

X1, X6, X57, X984, X1390

 

X1422

X1, X6, X57, X1413, X1422, X1436

 

X1432

X1, X6, X57, X893, X1431, X1432

 

X2006

X1, X6, X57, X1411, X2006, X2161

 

X2982

X1, X6, X57, X65, X1175, X2003, X2259, X2982

 

X2990

X1, X6, X57, X517, X2323, X2990

 

 

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.