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The Thomson cubic K002 and the Lemoine cubic K009 are both circum-cubics passing through the vertices of the medial triangle, X(3) and X(4). The tangent at X(3) passes through X(64) except for K009 which has a node at X(3).

They generate a pencil of cubics through these same 8 points and meeting the Euler line again at Q.

All these cubics are psK(Ω, X2, X3) with pseudo-pole Ω on the line X(6)-X(25) or, equivalently, psK(X6 x Q, X2, X3) where X6 x Q is a barycentric product. There is only one pK namely the Thomson cubic. See Pseudo-Pivotal Cubics and Poristic Triangles for further details.

See below for geometric properties of these cubics.

The following table shows a selection of these cubics according to the point Q on the Euler line.

Q

Ω

cubic

centers on the cubic : X3, X4, Q and

remark

X2

X6

K002

X1, X6, X9, X57, X223, X282, X1073, X1249

pK

X3

X184

K009

X32, X56, X1147

nodal cubic

X4

X25

K376

X64

 

X5

X51

K026

X5403, X5404

stelloid

X20

X154

K426

X3532

central cubic

X25

X1974

 

X206

 

X30

X1495

K446

X74, X133, X1511

circular cubic

X140

X6 x X140

K569

X54, X1493

 

X235

 

 

X2207, X2883

 

X297

X232

 

X1987

 

X403

 

 

X113

 

X426

 

 

X1092

 

X427

X1843

 

X66, X141, X3162

 

X429

 

 

X960

 

X439

 

 

X3053

 

X440

 

 

X71

 

X441

 

 

X248

 

X442

 

 

X10, X65, X942

 

X443

 

 

X2213

 

X468

 

 

X1177

 

X858

X2393

 

X67

 

X1312

 

 

X2575

 

X1313

 

 

X2574

 

X1368

 

 

X69

 

X1513

 

 

X511

 

X1529

 

 

X1503

 

X1532

 

 

X517, X1145, X1457

 

X1536

 

 

X516

 

X1551

 

 

X542

 

X1567

 

 

X2782

 

X1594

 

 

X1209

 

X2072

 

 

X265

 

X3079

 

 

X154

 

X3150

 

 

X879

 

X3154

 

 

X523

 

X6656

X1194

 

X695

 

The pencil contains seven other nodal cubics :

• three are decomposed into a sideline of ABC and a conic passing through X(3), X(4) and the remaining vertices of ABC and the medial triangle.

• four (not always real) with complicated corresponding points Q on the Euler line.

***

Properties of K = psK(X6 x Q, X2, X3) with Q on the Euler line

The cubic K = psK(X6 x Q, X2, X3) meets :

• the circumcircle again at the same points O1, O2, O3 as pK(X6, Q), a member of the Euler pencil, the remaining common points being obviously X(3), X(4) and Q.

• the line at infinity at the same points as pK(X6, aQ), another member of the Euler pencil, where aQ denotes the anticomplement of Q. The remaining common points lie on the Jerabek hyperbola namely A, B, C, X(3), X(4) and the isogonal conjugate Q4 of aQ.

K also passes through :

• Q1 = G-Ceva(X4 x Q), on the complement (H1) of the rectangular circum-hyperbola passing through X(20).

• Q2 = G-Ceva(X3 x Q), on the complement (H2) of the Jerabek hyperbola. Q2 is the tangential of Q.

• Q3 = X1073 x Q, on the line X(3)-X(64), the tangent at X(3) to K. Q3 is therefore the tangential of X(3).

Note the triads of collinear points on K :

X(3), Q2, Q4 - X(4), Q1, Q4 - Q, Q1, Q3.

 

See also Table 54 for a generalization.

table50