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The two most famous isotomic pivotal cubics K007 = Lucas cubic and K008 = Droussent cubic generate a pencil of cubics we call the Lucas - Droussent pencil.

Each member of this pencil is an isotomic pK with pivot on the line X4, X69 and isopivot on the Jerabek hyperbola.

These cubics are :

– the isogonal conjugates of pKs with pole X32, pivot on the Euler line. See Table 34.

– the complements of pKs with pole on the Brocard axis, pivot G i.e. the cubics of the pencil generated by K002 = Thomson cubic and K043 = Droussent medial cubic.

The nine base points of the pencil are A, B, C, G, H, X69, Ga, Gb, Gc (vertices of the antimedial triangle).

The following table presents a selection of these cubics according to the pivot P or the isopivot P* or the third point Q on the line X2, X69. The base points are omitted.

 

centers on the cubic

cubic

pivot P

X4

X193, X487, X488, X2996

K170

X69

X7, X8, X20, X189, X253, X329, X1032, X1034

K007 Lucas

X76

X6, X22, X76, X1670, X1671

K141

X264

X3, X254, X264, X1993

K045

X286

X21, X63, X72, X92, X286, X1441, X2997

 

X311

X54, X311, X1994, X2888

 

X314

X1, X65, X75, X81, X314, X321, X1764, X2995

K254

X315

X66, X315, X1370

 

X316

X67, X316, X524, X671, X858, X2373

K008 Droussent

X317

X68, X317

 

X340

X30, X265, X340, X1494, X2986

 

X511

X290, X385, X401, X511, X1916, X1972

K355

X621

X616, X621, X628, X2992

 

X622

X617, X622, X627, X2993

 

X877

X877, X879

 

X1232

X1173, X1232, X2889

 

X1234

X1175, X1234

 

X1235

X1176, X1235

 

X1236

X1177, X1236, X2892

 

X1330

X1330, X1654

 

X1843

X384, X1843

 

 

 

 

isopivot P*

X3

X3, X254, X264, X1993

 

X6

X6, X22, X76, X1670, X1671

K141

X54

X54, X311, X1994, X2888

 

X64

X64, X394, X2052

 

X65

X1, X65, X75, X81, X314, X321, X1764, X2995

K254

X66

X66, X315, X1370

 

X67

X67, X316, X524, X671, X858, X2373

K008 Droussent

X72

X21, X63, X72, X92, X286, X1441, X2997

 

X74

X74, X94, X146, X323, X1138, X1272, X2071

K279

X248

X248, X393, X1297, X2987

 

X265

X30, X265, X340, X1494, X2986

 

X290

X290, X385, X401, X511, X1916, X1972

K355

X1173

X1173, X1232, X2889

 

X1177

X1177, X1236, X2892

 

X1439

X1439, X1817, X1895

 

X1903

X1446, X1903, X2184, X2287

 

X2574

X1113, X2574, X2592

 

X2575

X1114, X2575, X2593

 

X2992

X616, X621, X628, X2992

 

 

 

 

point Q

X6

X6, X22, X76, X1670, X1671

K141

X81

X1, X65, X75, X81, X314, X321, X1764, X2995

K254

X141

X83, X141, X1369

 

X193

X193, X487, X488, X2996

 

X323

X74, X94, X146, X323, X1138, X1272, X2071

K279

X385

X290, X385, X401, X511, X1916, X1972

K355

X394

X64, X394, X2052K235

 

X524

X67, X316, X524, X671, X858, X2373

K008 Droussent

X1993

X3, X254, X264, X1993

K045

X1994

X54, X311, X1994, X2888

 

X2287

X1446, X1903, X2184, X2287