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Consider a pencil of rectangular hyperbolas generated by two distinct members, say (H1), (H2). The intersection Q of the polar lines of P in (H1), (H2) defines a quadratic involution f : P –> Q with fixed points the four common points of (H1), (H2) and singular points the vertices of the diagonal triangle of these four fixed points.

When (H1), (H2) are two diagonal conics (for instance the Stammler and Wallace hyperbolas), the four fixed points are the in/excenters of ABC and the diagonal triangle is ABC itself, a proper triangle, and f is the usual isogonal conjugation in ABC. The seven singular/fixed points are all finite and distinct.

When (H1), (H2) are two circum-conics (for instance the Jerabek and Kiepert hyperbolas), the four fixed points are A, B, C, X(4) and the diagonal triangle is the orthic triangle. f is the isogonal conjugation with respect to this latter triangle. Here again, the seven singular/fixed points are all finite and distinct.

In this page, we study several examples when the latter assertion is not necessarily entirely fulfilled i.e. when the diagonal triangle is not a proper triangle, having one vertex at infinity or two coincident vertices, etc.

See also Psi and Psi_P here.

 

The Jerabek-Stammler cubics JSpK

The Jerabek hyperbola (J) and the Stammler hyperbola (S) meet at two finite points (X3 and X6) and two infinite points (X2574 and X2575). The diagonal triangle has one vertex at infinity (namely X511) and two finite (always real) other vertices S1, S2 which are the common points of the Brocard circle and the line X(2), X(98), X(110), etc. These points also lie on the cubics K019, K048, K223, K417, K418, K907. They are now X(13414), X(13415) in ETC (2017-05-22).

For P distinct from the seven mentioned points, the cubic JSpK(P) is the locus of M such that P, M, JS(M) are collinear where JS hereby denotes the involution f defined above. Note that S1, S2 are the JS transforms of X(1114), X(1113) respectively. Moreover, the midpoint of M and JS(M) always lies on the Brocard axis and, for given X on the Brocard axis, the locus of M and JS(M) is the rectangular hyperbola with center X, passing through S1, S2, X2574, X2575.

The six cubics above are invariant under JS but K907 is the only JSpK. The others are nKs with respect to the diagonal triangle.

This cubic JSpK(P) must decompose when P lies on the Brocard axis, on the line at infinity, on a parallel at X(3) or X(6) to the asymptotes of (J).

JSpK(P) passes through X3, X6, X511, X2574, X2575, S1, S2 hence these cubics form a net. Moreover, JSpK(P) passes through P and Q = JS(P) which are analogous to the pivot and isopivot of usual pKs. Some other properties of pKs remain true :

• the polar conic of P is the rectangular hyperbola passing through P, X3, X6, X2574, X2575 hence JSpK(P) has two real asymptotes which are the parallels at P to those of the Jerabek hyperbola.

• the polar conic of Q is the conic passing through P, Q, X511, S1, S2 hence JSpK(P) has a third real asymptote which is the parallel at Q to the Brocard axis.

• JSpK(P), (J), (S) have four fixed common points X3, X6, X2574, X2575 hence each hyperbola must meet the cubic at two other points which lie on the polar line of P in the corresponding hyperbola. Recall that these two polar lines meet at Q = JS(P).

***

Further properties of JS

• Recall that JS has three singular points (X511, S1, S2) and four fixed points (X3, X6, X2574, X2575).

Note that all the points on a line passing through two singular points are transformed into the third singular point. In particular, the points on the Euler line of the 1st Brocard triangle are mapped onto X(511). This is the line passing through X(i) for i = 2, 98, 110, 114, 125, 147, 182, 184, 287, 1352, 1899, 1976, 2001, 3047, 3410, 3448, 3506, 4027, 5012, 5182, 5613, 5617, 5622, 5642, 5651, 5921, 5967, 5972, 5984, 5985, 5986, 5987, 6036, 6054, 6055, 6230, 6231, 6721, 6723, 6770, 6771, 6773, 6774, 6776, 8593, 9140, 9143, 9306, 9544, 9744, 9759, 9775, 10168, 10334, 10352, 10353, 11003, 11005, 11161, 11177, 11178, 11179, 11180, 11442, 11579, 11653, 12177, 12192, 12827, 13029, 13031, 13198.

Similarly, the (non singular) points on the line X511, S1 (resp. X511, S2) are mapped onto S2 (resp. S1), red and green points in the list below.

• JS coincide with isogonal conjugation in ABC for any point on K019.

• more generally, if X is a point on the Brocard axis, then JS coincide with the isoconjugation with pole X for any point on nK0(X, X647).

• JS coincide with the reflection in the Brocard axis for any point on the Brocard circle which is therefore (globally) invariant under JS.

• JS coincide with the inversion in the Brocard circle for any point on the Brocard axis which is therefore also (globally) invariant under JS.

• as already mentioned above, for any (finite) X on the Brocard axis, JS coincide with the reflection in X for any point that lies on the rectangular hyperbola with center X, passing through S1, S2, X2574, X2575. When X is X(3) or X(6), these hyperbolas split into two perpendicular lines.

***

A list of pairs {P, JS(P)} (excluding the singular cases above) by Peter Moses (2017-05-31) :

{1,1046}, {3,3}, {4,185}, {5,6102}, {6,6}, {15,16}, {20,5907}, {30,5663}, {32,39}, {40,8235}, {50,566}, {52,569}, {58,386}, {61,62}, {69,6467}, {74,1495}, {187,574}, {194,3491}, {216,577}, {284,579}, {323,9976}, {371,372}, {373,1992}, {376,5650}, {389,578}, {500,582}, {511,542}, {512,690}, {513,8674}, {514,2774}, {515,2779}, {516,2772}, {517,2771}, {518,2836}, {519,2842}, {520,9033}, {521,2850}, {522,2773}, {523,526}, {524,2854}, {525,9517}, {567,568}, {570,571}, {572,573}, {575,576}, {580,581}, {583,584}, {800,5065}, {895,3292}, {970,13323}, {974,11064}, {991,13329}, {1030,5124}, {1113,13415}, {1114,13414}, {1151,1152}, {1333,4261}, {1340,1380}, {1341,1379}, {1342,1670}, {1343,1671}, {1350,5085}, {1351,5050}, {1384,5024}, {1499,2780}, {1503,2781}, {1504,5062}, {1505,5058}, {1578,1579}, {1662,1664}, {1663,1665}, {1666,1668}, {1667,1669}, {1685,13333}, {1686,13332}, {1687,1689}, {1688,1690}, {1691,3094}, {1692,5028}, {2012,13324}, {2076,5116}, {2080,11171}, {2092,5019}, {2104,13414}, {2105,13415}, {2220,5069}, {2245,2278}, {2271,5021}, {2305,5110}, {2558,13325}, {2559,13326}, {2560,2561}, {2562,13328}, {2563,13327}, {2574,2574},{2575,2575}, {2673,2674}, {2775,3309}, {2776,3667}, {2777,6000}, {2778,6001}, {2965,13351}, {3003,5063}, {3053,5013}, {3095,3398}, {3098,5092}, {3111,5118}, {3284,5158}, {3285,4286}, {3286,5132}, {3311,3312}, {3313,5157}, {3364,3390}, {3365,3389}, {3368,3395}, {3369,3396}, {3371,3386}, {3372,3385}, {3379,3393}, {3380,3394}, {3592,3594}, {3736,5156}, {4251,4253}, {4252,4255}, {4254,5120}, {4256,4257}, {4258,5022}, {4259,5135}, {4260,5138}, {4262,5030}, {4263,5042}, {4264,5105}, {4265,5096}, {4266,5053}, {4268,4271}, {4272,5115}, {4273,5165}, {4274,5114}, {4275,5153}, {4277,5035}, {4279,5145}, {4283,5009}, {4284,5037}, {4287,5036}, {4289,5043}, {4290,5109}, {5007,7772}, {5034,5052}, {5038,13330}, {5171,13334}, {5237,5238}, {5351,5352}, {5396,5398}, {5421,13345}, {5668,5669}, {6090,10602}, {6199,6395}, {6200,6396}, {6221,6398}, {6243,13353}, {6407,6408}, {6409,6410}, {6411,6412}, {6417,6418}, {6419,6420}, {6421,6424}, {6422,6423}, {6425,6426}, {6427,6428}, {6429,6430}, {6431,6432}, {6433,6434}, {6435,6436}, {6437,6438}, {6439,6440}, {6441,6442}, {6445,6446}, {6447,6448}, {6449,6450}, {6451,6452}, {6453,6454}, {6455,6456}, {6468,6469}, {6470,6471}, {6472,6473}, {6474,6475}, {6476,6477}, {6478,6479}, {6480,6481}, {6482,6483}, {6484,6485}, {6486,6487}, {6488,6489}, {6490,6491}, {6492,6493}, {6494,6495}, {6496,6497}, {6498,6499}, {6500,6501}, {6519,6522}, {8115,13414}, {8116,13415}, {8586,10485}, {8588,8589}, {8675,9003}, {9729,13346}, {9730,13352}, {9735,13349}, {9736,13350}, {9737,13335}, {9786,11425}, {9821,12054}, {10137,10138}, {10139,10140}, {10141,10142}, {10143,10144}, {10145,10146}, {10147,10148}, {10625,13336}, {10634,10635}, {10645,10646}, {10897,10898}, {11426,11432}, {11430,11438}, {11480,11481}, {11485,11486}, {11513,11514}, {11515,11516}, {12050,12051}, {12212,13331}, {13337,13338}, {13339,13340}, {13341,13342}, {13343,13344}, {13347,13348}, {13354,13355}, {13356,13357}.

***

Cubics

A circum-cubic invariant under JS must contain X(511), S1, S2 and the JS transforms of A, B, C which are the traces of the trilinear polar of X(647) on the sidelines of ABC. Such cubic must be a nK0(Ω, X647) with Ω on the Brocard axis hence it belongs to a pencil of cubics. The cubic passing through a given point M must also contain JS(M) and then Ω is the barycentric product M x JS(M). The most remarkable are K019 = nK0(X6, X647) and K223 = nK0(X32, X647) = cK(#X6, X647). Other example :

K908 = nK0(X577, X647) = cK(#X3, X647) passing through X(3), X(450), X(511), X(895), X(3292), X(13414), X(13415).

***

The table below presents a selection of cubics JSpK(P) passing through at least eleven ETC centers (X3, X6, X511, X2574, X2575, S1 = X13414, S2 = X13415 are not repeated).

K907 = JSpK(X51) is one of the most remarkable, passing through several common centers. Those highlighted in yellow are the most "prolific" in ETC centers. This table was built with contributions by Peter Moses.

P

X(i) on JSpK(P) for i =

 

P

X(i) on JSpK(P) for i =

4

4, 155, 185, 1352

3917

2, 20, 69, 154, 3917, 5907, 6467

20

20, 1498, 5907, 6776

4176

69, 1611, 4176, 6467

22

22, 159, 184, 12163

5250

1, 405, 1046, 5250

23

23, 110, 2930, 5888, 10620

5448

5, 5448, 6102, 12084

31

1, 31, 1046, 1740, 1754

5449

5, 26, 5449, 6102, 8548

38

1, 38, 1046, 1764

5462

5, 1353, 1493, 5462, 6102

51

2, 4, 51, 185, 193, 3167

5640

2, 381, 5640, 9716

63

1, 63, 610, 1046

5876

5, 1657, 5876, 6102, 12316

69

69, 159, 1352, 6467

5878

4, 185, 5878, 12085

74

74, 1495, 11472, 11579

5890

4, 185, 381, 3431, 5890, 11179

146

146, 399, 2935, 3818

5907

5, 20, 1352, 5907, 6102

186

54, 74, 186, 1495, 2931, 5621, 12584

5943

2, 5, 3629, 5943, 6102

305

69, 305, 1613, 6467

5946

5, 381, 5946, 6102, 10168

323

110, 323, 399, 5643, 9976

6090

154, 6090, 9306, 10602

340

340, 1352, 2992, 2993

6193

155, 5562, 6193, 9937

394

394, 1498, 6391, 9306

6225

4, 185, 1498, 6225

612

40, 612, 940, 8235

6241

4, 185, 382, 6241, 11270, 12163

774

1, 774, 1046, 1715

6524

4, 185, 6524, 6617

1205

67, 1177, 1205, 11579

6636

2916, 5012, 6636, 12307

1368

141, 394, 1368, 2883

6644

1147, 4550, 6644, 10605

1495

74, 110, 1113, 1114, 1495

7998

2, 599, 3534, 7712, 7998

1531

265, 1352, 1531, 4846

9140

23, 323, 381, 599, 9140, 9976

1619

25, 159, 1619, 12085

9143

399, 2930, 5640, 7998, 9143

1621

1, 1046, 1621, 8053

9781

4, 185, 3526, 9781

1899

4, 25, 69, 185, 394, 1899, 6467

10110

4, 140, 185, 1173, 10110, 12007

1931

1, 1046, 1931, 9509

11064

974, 2935, 5972, 11064

1962

1, 165, 1046, 1962

11206

51, 159, 1498, 3819, 11206

2052

4, 185, 2052, 6638

11381

4, 20, 64, 185, 5907, 5921, 11381, 12164

2292

1, 40, 1046, 2292, 8235

11433

4, 185, 5020, 11433

2650

1, 1046, 2650, 3751

11441

155, 1204, 1498, 11441

2888

195, 2888, 2917, 5889

11444

20, 1656, 5907, 11444

3060

2, 382, 3060, 6144

11455

4, 185, 3534, 11455, 11738

3193

1, 155, 1046, 3193

11457

4, 185, 7517, 11457

3292

110, 895, 3292, 8115, 8116

11550

4, 22, 66, 185, 11442, 11550, 12220

3491

194, 1352, 3491, 4048

11572

4, 185, 6145, 7488, 11572

3567

4, 185, 1656, 3567

12081

1, 1046, 2077, 12081

3743

1, 1046, 1386, 3579, 3743

12112

74, 399, 1495, 12112

3747

1, 238, 1046, 1742, 3747, 9441

12233

5, 1593, 6102, 12233

3794

1, 2, 1046, 3794

12290

4, 185, 1657, 12290

3819

2, 141, 550, 3819, 6030

12359

5, 6102, 7387, 12359

3868

1, 1046, 3868, 10441

12383

399, 2931, 11459, 12383

3869

1, 1046, 3869, 10477

12824

381, 858, 5642, 12824

3881

1, 1046, 3881, 5482

 

 

3915

1, 1046, 1724, 3915

 

 

Remarks :

• when P lies on the line passing through X(1), X(21), X(31), X(38), X(47), X(58), X(63), X(81), etc, the cubic JSpK(P) passes through X1 and X1046 = JS(X1).

• when P lies on the line passing through X(4), X(51), X(185), X(389), etc, the cubic JSpK(P) passes through X4 and X185 = JS(X4).

 

The Kiepert-Wallace cubics KWpK

The Kiepert and Wallace hyperbolas meet at X(2) twice (since the tangent is the same namely the line X2, X6) and two real infinite points (namely X3413, X3414). Recall that the Wallace hyperbola (W) is the anticomplement of the Kiepert hyperbola (K).

The diagonal triangle of these four points has two coincident vertices at X(2) and the third vertex is X(524) at infinity.

For P distinct from the previous mentioned points, the cubic KWpK(P) is the locus of M such that P, M, KW(M) are collinear where KW hereby denotes the involution f defined at the top of this page. Moreover, the midpoint of M and KW(M) always lies on the line passing through X(2) and X(6).

KWpK(P) is a nodal cubic with node X(2) and passes through X(524), X(3413), X(3414).

The polar conic of P passes through X(2), X(3413), X(3414), P and is tangent at X(2) to the line X(2)X(6).

The polar conic of Q = KW(P) passes through X(2), X(524), P, Q and is tangent at X(2) to the line X(2)X(99).

KWpK(P) meets (K) [resp. (W)] at X(2) counted twice, X(3413), X(3414) and two other points K1, K2 [resp. W1, W2] on the polar line of P in (K) [resp. (W)]. These two polar lines meet at Q.

***

Further properties of KW

• Recall that KW has three singular points (X2 counted twice and X524) and two fixed points (X3413, X3414).

Every (non singular) point on the line X(2)X(6) has its KW image at X(2) and every (non singular) point on the line X(2)X(99) has its KW image at X(524).

Every (non singular) point on the line at infinity has its KW image also on the line at infinity. The lines passing through X(2) and these two infinite points are symmetric in the lines X(2)X(3413), X(2)X(341) i.e. the axes of the Steiner ellipses.

• KW coincide with isogonal conjugation in ABC for any point on K018 = nK0(X6, X523).

• KW coincide with isotomic conjugation in ABC for any point on K185 = nK0(X2, X523) = cK(#X2, X523).

• more generally, if X is a point on the line X(2)X(6), then KW coincide with the isoconjugation with pole X for any point on nK0(X, X523). See K205 (X = X1989) and K381 (X = X32) for instance.

• KW coincide with the reflection in X(2) for any point on the axes of the Steiner ellipse. In particular, KW swaps the foci of the Steiner inellipse.

***

A list of pairs {P, KW(P)} (excluding the singular cases above) by Peter Moses (2017-05-31) :

{3,1352}, {4,6776}, {5,182}, {8,9791}, {13,15}, {14,16}, {20,5921}, {22,11442}, {23,3448}, {25,1899}, {30,542}, {76,3094}, {98,1513}, {110,858}, {125,468}, {147,5999}, {184,427}, {239,6651}, {287,297}, {305,3981}, {315,4048}, {376,11180}, {381,11179}, {383,6773}, {403,5622}, {511,2782}, {512,804}, {513,2787}, {514,2786}, {515,2792}, {516,2784}, {517,2783}, {518,2795}, {519,2796}, {520,2797}, {521,2798}, {522,2785}, {523,690}, {525,2799}, {530,531}, {538,5969}, {547,10168}, {549,11178}, {616,621}, {617,622}, {618,623}, {619,624}, {694,3978}, {826,9479}, {868,5967}, {1080,6770}, {1194,4074}, {1312,13415}, {1313,13414}, {1368,9306}, {1369,10328}, {1499,2793}, {1503,2794}, {1689,1690}, {1916,9865}, {1976,2450}, {2788,3309}, {2789,3667}, {2790,6000}, {2791,6001}, {3120,4062}, {3124,3266}, {3410,6636}, {3413,3413}, {3414,3414}, {3642,3643}, {3849,9830}, {3934,10007}, {5012,5133}, {5028,7789}, {5116,7785}, {5159,5972}, {5169,11003}, {5978,5979}, {6032,10166}, {6034,7799}, {6036,10011}, {6108,6109}, {6536,8013}, {6542,6650}, {6669,6671}, {6670,6672}, {7426,9140}, {8290,9866}, {8352,8593}, {8598,11161}, {8878,10329}, {9143,10989}, {10160,10162}, {10653,10654}, {11078,11092}, {11579,11799}.

***

Cubics

A circum-cubic invariant under KW must contain X(2), X(524) and the KW transforms of A, B, C which are the traces of the trilinear polar of X(523) on the sidelines of ABC. Such cubic must be a nK0(Ω, X523) with Ω on the line X(2)X(6) hence it belongs to a pencil of cubics. The cubic passing through a given point M must also contain KW(M) and then Ω is the barycentric product M x KW(M). The most remarkable are K018 = nK0(X6, X523) and K185 = nK0(X2, X523) = cK(#X2, X523). The cubic nK0(X1648, X523) splits into the trilinear polar of X(523) and the circum-conic with perspector X(690). Other examples :

nK0(X394, X523) passing through X2, X394, X524, X2987

nK0(X1641, X523) passing through X2, X524, X543, X1641

nK0(X2086, X523) passing through X2, X512, X524, X804, X2086

nK0(X3051, X523) passing through X2, X184, X237, X427, X524, X3051

***

The most remarkable examples are probably K801 = KWpK(X5) and K906 = KWpK(X3). See light blue cells in the table below which presents a selection of cubics KWpK(P) passing through at least nine ETC centers (X2, X524, X3413, X3414 are not repeated). Those highlighted in yellow are the most "prolific" in ETC centers. It was built with contributions by Peter Moses.

P

X(i) on KWpK(P) for i =

3

3, 485, 486, 1352, 1689, 1690, 7618

4

4, 487, 488, 6776, 7620

5

5, 182, 627, 628, 641, 642, 7617, 8149, 8150, 11261

15

13, 15, 18, 1689, 1690, 9885

16

14, 16, 17, 1689, 1690, 9886

32

32, 1676, 1677, 1689, 1690

39

39, 1689, 1690, 5403, 5404

61

61, 1689, 1690, 3391, 3392

62

62, 1689, 1690, 3366, 3367

140

17, 18, 140, 6118, 6119, 7619

182

5, 182, 1689, 1690, 7622

187

187, 1689, 1690, 2482, 6108, 6109, 6292

265

265, 11078, 11092, 11579, 11799

316

83, 316, 671, 5978, 5979

371

371, 1689, 1690, 3387, 3388

372

372, 1689, 1690, 3373, 3374

575

546, 575, 1689, 1690, 7619

576

576, 1689, 1690, 3627, 7617

1340

1340, 1349, 1689, 1690, 2543, 5639

1341

1341, 1348, 1689, 1690, 2542, 5638

1348

5, 182, 1341, 1348, 2542, 3557, 5638, 6177

1349

5, 182, 1340, 1349, 2543, 3558, 5639, 6178

1350

20, 1350, 1689, 1690, 5921, 6194

1351

382, 1351, 1689, 1690, 7615

1550

98, 1513, 1550, 11078, 11092

1691

83, 98, 1513, 1689, 1690, 1691

1975

20, 194, 637, 638, 1975, 5921

2076

1689, 1690, 2076, 2896, 8782, 11676

2092

10, 1689, 1690, 2051, 2092

3094

76, 262, 1689, 1690, 3094

3098

550, 1689, 1690, 3098, 3642, 3643

3369

1689, 1690, 3369, 5401, 5402

P

X(i) on KWpK(P) for i =

3380

1140, 1689, 1690, 3370, 3380

3394

1139, 1689, 1690, 3394, 3397

3396

1689, 1690, 3381, 3382, 3396

3581

1689, 1690, 3581, 11078, 11092

3818

5, 182, 3642, 3643, 3818, 5569

4048

3, 194, 315, 1352, 4048

5050

381, 1689, 1690, 5050, 11179

5103

83, 1916, 5103, 9166, 9865, 12177

5254

4, 76, 371, 372, 3094, 5254, 6776

5318

4, 13, 15, 5318, 6776

5321

4, 14, 16, 5321, 6776

5480

4, 262, 5480, 6309, 6776, 7615

6199

1327, 1689, 1690, 6199, 10194

6390

39, 2482, 5978, 5979, 6390

6395

1328, 1689, 1690, 6395, 10195

6425

1131, 1689, 1690, 3317, 6425

6426

1132, 1689, 1690, 3316, 6426

6560

485, 1328, 6560, 10653, 10654

6561

486, 1327, 6561, 10653, 10654

6656

76, 83, 1342, 1343, 3094, 6656

7789

3, 39, 639, 640, 1352, 5028, 7789

8550

4, 6776, 7607, 7618, 8550

8586

671, 1689, 1690, 7608, 8586

9732

487, 1689, 1690, 9732, 12222

9733

488, 1689, 1690, 9733, 12221

10485

598, 1689, 1690, 7607, 10485

11178

549, 3642, 3643, 7617, 11178

11179

381, 7618, 10653, 10654, 11179

11477

1689, 1690, 3146, 7620, 11477

13331

1689, 1690, 7757, 7812, 13331

 

 

 

 

Remark : the very frequent occurence of the pair X(1689), X(1690) corresponds to P on the Brocard axis. These two points (on the Brocard axis) are swapped under KW.