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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 and other related topics in pages Q149 and K1191.

The examples mentioned below are :

• The Jerabek-Stammler cubics JSpK • The Kiepert-Wallace cubics KWpK • The Feuerbach cubics FpK

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, K1106. 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 and K1106 are 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) which may be seen as the in/excenters of the non proper triangle with vertices X511, S1, S2.

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) and K1085 = JSpK(X3917) are two remarkable examples, 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.

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.

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.

The Feuerbach cubics FpK

Under the symbolic substitution SS{a → √a}, the Jerabek hyperbola is transformed into the Feuerbach hyperbola (F) and the Stammler hyperbola is transformed into the diagonal rectangular hyperbola (F') passing through X(1), X(9), X(40), X(188), X(191), X(366), X(1045), X(1050), X(1490), X(2136), X(2949), X(2950), X(2951), X(3174), X(3307), X(3308), etc. (F') is the Jerabek hyperbola of the excentral triangle.

(F) and (F') pass through X(1), X(9) and two points X(3307), X(3308) on the line at infinity. The diagonal triangle (T) of the quadrilateral formed with these four points has vertices X(518) at infinity and X(24646), X(24647).

As above, an isogonal conjugation F (the involution f defined at the top of this page) in the non proper triangle (T) is defined and for any P not lying on the sidelines of (T), the cubic FpK(P) is the locus of M such that P, M, F(M) are collinear.

Note that F coincide with the usual isogonal conjugation in ABC for any point M on the Pelletier strophoid K040 which must contain X(518), X(24646), X(24647).

Every FpK(P) passes through these three latter points and also X(1), X(9), X(3307), X(3308) which are in a way the in/excenters of (T). Obviously, FpK(P) passes through P, F(P), the vertices of the cevian triangle of P in (T).

The most remarkable examples are K1083 (P = X354) and K1084 (P = X210), these pivots being symmetric about X(2).