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Let P = u : v : w be a point and pK(P) = pK(X6, P) the isogonal pivotal cubic with pivot P.

pK(P) meets the circumcircle (O) at A, B, C and three other points Q1, Q2, Q3 which are not necessarily all real nor distinct. See here for further informations. Denote by T the (possibly improper) triangle with vertices Q1, Q2, Q3.

The construction of Q1, Q2, Q3 is not ruler and compass possible. It needs to intersect (O) and another conic. One solution is the homothetic under h(P, 1/2) of the polar conic of P in pK(P), a diagonal conic passing through P and the in/excenters of ABC, discarding the image of the center of the diagonal conic which is the trilinear pole of the line through X(6) and the barycentric square of P.

For example, with P = X(2), X(3), X(6), T is the Thomson triangle, CircumNormal triangle, Grebe triangle respectively. These are proper and acute triangles for any reference triangle ABC. With P = X(1), T is the circumcevian triangle of X(1).

Obviously O is the circumcenter of T and P is its orthocenter. The polar circle of T is that with center P and radius :

-((c^2 u v+b^2 u w+a^2 v w)/(2 (u+v+w)^2)) hence T is acutangle for P inside (O).

Recall that T is a proper triangle if and only if P is (strictly) inside aH3, the anticomplement of the Steiner deltoid, which is the case when P is (strictly) inside (O).

Let ΦP be the isogonal conjugation with respect to T. ΦP swaps the vertices of ABC and the infinite points of its altitudes, also O and P. ΦP obviously fixes the in/excenters of T.

It follows that an isogonal pivotal cubic with respect to T which is also a circumcubic in ABC must have its pivot at H and therefore passes through the vertices of the cevian triangle H1H2H3 of H in Q1Q2Q3.

Hence, any pK(P) = pK(X6, P) is associated with a unique cubic K(P) we call the sister of pK(P). In other words, K(P) is the pivotal cubic with pivot H which is invariant under ΦP, the isogonal conjugation with respect to T.

K(P) contains seven fixed points namely A, B, C, H and the infinite points of the altitudes of ABC hence these cubics belong to a same net of cubics with asymptotes La, Lb, Lc perpendicular to the sidelines of ABC or parallel to those of the Darboux cubic K004.

table58KP

K(P) also contains :

• Q1, Q2, Q3,

• the in/excenters of T (not represented),

• the ΦP-image Z of H which is the isopivot hence the common tangential of H, Q1, Q2, Q3 in K(P). This point has first barycentric coordinate :

a^2 (a^4 u-a^2 b^2 u-a^2 c^2 u+a^4 v-a^2 b^2 v+b^2 c^2 v-c^4 v+a^4 w-b^4 w-a^2 c^2 w+b^2 c^2 w), and can be construed as the Ceva conjugate of X(3) x tcP and X(6).

See ETC pairs {P, Z} and {Z, P} at the bottom of this page.

The mapping P –> Z is affine with fixed points X(110), X(2574), X(2575). See the related cubic K839.

It can easily be verified that K(P) contains the vertices Za, Zb, Zc of the pedal triangle of Z which makes already 21 clearly identified points on K(P).

K(P) meets pK(P) at six points on (O) and three other (not always all real) collinear points R1, R2, R3 on a line Lo passing through O.

K(P) meets K004 at three points at infinity, A, B, C, H hence two other (not always real) points S1, S2 which lie on a line passing through Z.

***

The orthic line of K(P) is the Euler line for every P. This means that the polar conic of any point M on the Euler line with respect to any cubic K(P) is a rectangular hyperbola. This is in particular the case of H (which lies on every K(P)).

The orthic line of the hessian H(P) of K(P) passes through H. It coincides with the Euler line if and only if P lies on a rectangular hyperbola passing through X(4), X(20), X(2574), X(2575), X(7691). Further details and developments are available in the paper Bi-isogonal and Tri-isogonal Pivotal Cubics.

 

Special cubics K(P)

Since the infinite points are those of the altitudes of ABC, K(P) cannot be circular nor equilateral unless its decomposes.

• K(P) passes through P if and only if either :

– P lies on the line at infinity in which case it must decompose since it has already three known points at infinity. This is excluded in the sequel.

– P lies on (O) in which case K(P) meets (O) again at two points lying on the parallel at O to the Simson line of P. See K839 for instance.

– P lies on the Euler. See below.

• K(P) is a K0 (and at the same time a K+) if and only if P lies on the line passing through X20, X185, X193, X194, X511, X3164, X5889 etc. When P = X20, K(P) is the Darboux cubic (which is therefore its own sister) and when P = X5889, K(P) splits into the altitudes of ABC. These are the only cases for which K(P) is a K++.

The asymptotes concur at X on the Euler line and Z lies on the line X(4), X(6). See K840 for instance.

• K(P) is a psK (or a pK) if and only if P lies on a central cubic with center H, passing through X20, X3146. See K841 for instance and obviously also K004.

• K(P) is a nK (or a nK0) if and only if P lies on the line at infinity (see above) or on a conic with center H passing through X110, X1897.

In this latter case, Z lies on (O) hence Za, Zb, Zc are collinear on the Simson line of Z. The pole of the nK is the barycentric product H x Z and lies on the circum-conic with perspector X(25), passing through X(112), X(648), X(1783). The root is the tripole of the Simson line of Z hence it lies on the Simson cubic K010. K(P) meets (O) at Z and two other points on the Steiner line of Z. See K839 and K842 for instance.

There are two nK0s of this type obtained when Z is an intersection of (O) and the line X(4), X(6). These points and the corresponding points P are complicated and unlisted in ETC.

 

Selected examples

The table below shows a selection of cubics pK(P) and their sisters K(P). This table was built with contributions by Peter Moses.

Recall that K(P) is always a circumcubic passing through H and having three real asymptotes perpendicular to the sidelines of ABC.

P

pK(P)

K(P)

Z

Xi on K(P) for i =

remarks

X1

bisectors

K838

Z1

4

T is the circumcevian triangle of X1, TCCT 6.22

X2

K002

K615

X154

2, 3, 4, 64, 154, 3424, 5373

T is the Thomson triangle, K(P) is spK(X20, X376)

X3

K003

K405

X6759

3, 4, 64, 1676, 1677, 6759

T is the CircumNormal triangle, K(P) is spK(X20, X550)

X4

K006

K376

X3

3, 4, 64, 6523

K(P) is psK(X25, X2, X3) and spK(X20, X3)

X5

K005

K848

X10282

3, 4, 5, 64, 10282

K(P) is spK(X20, X548)

X6

K102

K642

X206

4, 206, 1676, 1677

T is the Grebe triangle

X20

K004

K004

X1498

see the page K004

T is the circumcevian triangle of X3, TCCT 6.12

X27

K109

 

X2328

3, 4, 27, 64, 2328

 

X99

K035

 

 

4, 99, 1379, 1380

 

X110

K316

K839

X110

4, 110, 1113, 1114

K(P) is nK(X112, X2407, X4)

X185

 

K840

X2883

4, 2883

K(P) is a K0+

X193

 

 

X6

4, 6

K(P) is a K0+

X194

 

 

 

4

K(P) is a K0+

X382

 

K846

X3357

3, 4, 64, 382, 3357

K(P) is spK(X20, X5)

X476

K130

 

 

4, 74, 110, 476

 

X631

K832

 

Z631

3, 4, 64, 631

 

X1351

 

K843

X182

4, 182, 1344, 1345, 1676, 1677

 

X1897

 

K842

X109

4, 109

K(P) is nK(X6 x X653, X2406, X4)

X3146

 

K841

X64

3, 4, 64, 3146

K(P) is psK(X64, X253, X4) and spK(X20, X4)

X3149

 

K844

 

3, 4, 56, 64, 945, 3149, 5687, 10306

 

X3164

 

 

 

4

K(P) is a K0+

X3543

 

K847

X10606

3, 4, 64, 10606

K(P) is spK(X20, X2)

X3580

 

 

X1495

4, 74, 1495, 5000, 5001

See the related Table 55

X9965

 

K845

X2192

1, 4, 57, 84, 1750, 2192, 10306

X9965 = aX329

P127

 

K127

 

4, 5, 3146

K(P) is a K0+

P820

 

K820

 

3, 4, 64, 182, 1350

 

Notes :

Z1 lies on the lines {X1,X1437}, {X6,X2333}, {X31,X56}, {X40,X692}, {X47,X859}, {X65,X184}, etc, SEARCH = -1.20708042684458.

Z631 lies on the lines {X3,X64}, {X25,X1614}, {X49,X1660}, etc, SEARCH = -3.15562157454102.

Any cubic K(P) with P on the Brocard axis passes through X(1676), X(1677).

 

Cubics K(P) with P on the Euler line

When P lies on the Euler line, K(P) has a lot of specific properties. See the green cells in the table above.

table58euler

• P lies on K(P) as said above.

• K(P) passes through X(3) and X(64).

• K(P) is a member of the pencil generated by K004 and the union of the line at infinity and the Jerabek hyperbola. More precisely, K(P) is spK(X20, midpoint X20-P) as in CL055. See also Table 54, column P = X20.

• K(P) is also a member of the pencil F(P) generated by pK(P) and the union of the Euler line and the circumcircle. See a study of this pencil below.

• The polar conic of P is a rectangular hyperbola.

• Z lies on the line X3, X64.

• The sidelines of T envelope the parabola (P) which is the reflection of the Kiepert parabola about O.

• The in/excenters Io, I1, I2, I3 of T, which lie on K(P), also lie on (H), the reflection of the Stammler rectangular hyperbola about O.

• The Lemoine point KT of T also lie on (H).

 

Pencils pK(P), (O) U (E), K(P) with P on the Euler line

As said above, when P lies on the Euler line, K(P) belongs to the pencil F(P) generated by pK(P) and the union of the Euler line (E) and the circumcircle (O). It is clear that, for a given P, all the cubics of F(P) pass through X3, X4, P on (E) and meet (O) at the same six points.

Each cubic of F(P) is spK(Q, D = midpoint P-Q) for some Q on the Euler line, meeting the line at infinity at the same points as pK(X6, Q) and passing through the foci of the inconic with center D.

This pencil F(P) contains a good number of remarkable other cubics, most of them already mentioned in Table 54, although organized differently.

This also allows a connection with the M-OAP points of Table 53. Indeed, F(P) always contains :

• a cubic passing through the X3-OAP points,

• a cubic passing through the isogonal conjugates of the X3-OAP points,

• a cubic M(P) passing through the M-OAP points where M is the reflection of P about X3.

These three cubics are not always all distinct.

Let S be the abscissa of P in (O = X3, G = X2) i.e. OP = S OG (vectors). Consider the point Ps with abscissa (–3 – 2S) in (O, G) or, equivalently, such that X(20)Ps = –2 OP. M(P) is then spK(Ps, midpoint P-Ps) as in CL055.

The following table gives a selection of these F(P), each being represented by one column. Each line also represents a pencil of cubics which share a common property.

P

X30

X2

X3

X20

X5

X4

X376

X381

X382

Q

notes

pK(X6, P)

K001

K002

K003

K004

K005

K006

K243

 

 

P

 

K(P) / ∞K004

(L)U(J)

K615

K405

K004

K848

K376

 

 

K846

X20

(L1)

S(P) / ∞K003

(L)U(J)

K581

K003

K080

K026

K028

K309

K358

K525

X3

(L2)

psK1

K447

K002

K361

K443

K849

K028

 

 

 

aD

(L3)

psK2

K446

K002

K009

K426

K026

K376

 

 

 

aaD

 

X3-OAP

K811

K810

K405

K814

K026

K006

 

K762

 

S

(L4)

X3-OAP*

K854

K759

K361

K080

 

K006

K851

 

 

S'

 

M

X30

X376

X3

X4

X550

X20

X2

X3534

X1657

 

 

M(P)

K446

 

K405

K443

 

K855

K243

 

 

Ps

(L5)

∞K002

(L)U(J)

K002

 

K047

 

 

K851

K762

 

X2

 

∞K005

(L)U(J)

 

K361

K566

K005

 

 

 

 

X5

 

∞K006

(L)U(J)

K804

K009

K443

 

K006

 

 

 

X4

 

F(X2)

K187

K002

 

K706

 

 

 

K358

 

s(X2)

(L6)

F(X3)

K187

K810

K003

K443

 

K376

K851

 

 

s(X3)

 

F(X4)

K187

 

 

K426

 

K006

 

 

K525

s(X4)

 

F(X5)

K187

K759

K009

K850

K005

K028

 

K762

K846

s(X5)

(L7)

F(X20)

K187

 

 

K004

 

K855

 

 

 

s(X20)

 

F(X376)

K187

K615

 

K047

 

 

 

 

 

s(X376)

 

notes

(C1)

(C2)

(C3)

(C4)

 

 

 

 

 

 

 

Notations and comments

K(P) is pK(X6, P)'s sister and S(P) is a stelloid with asymptotes parallel to those of the McCay cubic K003.

S and S' are the images of D under the homotheties with center X4, ratios 4/3 and 2/3 respectively.

(L)U(J) is the union of the line at infinity and the Jerabek hyperbola.

In the line ∞Knnn, all the cubics pass through the infinite points of Knnn.

In the line F(Xi), all the cubics pass through the foci of the inconic with center Xi and s(Xi) is the reflection of P about Xi.

***

Notes

• (L1) : the pencil of Euler's sisters also contains K820, K841, K844, K847. This column P = X20 in Table 54.

• (L2) : the pencil of McCay stelloids also contains K665, K852.

• (L3) : the pencil of psK1s also contains K813, K841.

• (L4) : the pencil of X3-OAP cubics also contains K812, K813.

• (L5) : the pencil of M-OAP cubics also contains the stelloid K665 (M = X5), K855 (M = X20).

• (L6) : the pencil of cubics passing through the foci of the Steiner inellipse also contains K812, K847.

• (L7) : the pencil of cubics passing through the foci of the MacBeath inconic have the same tangents at X3, X4 passing through X74 except when P = X3, X4 since the corresponding cubics K009, K028 are nodal.

 

• (C1) : all the cubics in this column are circular and pass through X(30), X(74).

• (C2) : all the cubics in this column pass through X(2) and the vertices of the Thomson triangle.

• (C3) : all the cubics in this column are CircumNormal cubics. Additional cubic : K664.

• (C4) : all the cubics in this column are central with center O.

 

ETC pairs {P, Z} and {Z, P}

Here are lists of pairs {P, Z} and {Z, P} computed by Peter Moses, updated 2016-11-02.

Pairs {P, Z}

{2,154}, {3,6759}, {4,3}, {5,10282}, {6,206}, {8,3556}, {20,1498}, {24,10539}, {25,9306}, {27,2328}, {29,2360}, {30,6000}, {51,10192}, {52,5}, {63,10537}, {68,26}, {69,159}, {74,9934}, {92,3185}, {110,110}, {113,1511}, {144,3197}, {145,221}, {146,2935}, {155,1147}, {185,2883}, {186,10540}, {193,6}, {195,10274}, {264,160}, {317,157}, {324,418}, {382,3357}, {385,1971}, {394,1660}, {428,3819}, {511,1503}, {512,3566}, {514,8676}, {517,6001}, {518,3827}, {519,2390}, {520,523}, {521,513}, {523,924}, {524,2393}, {525,512}, {539,1154}, {542,2781}, {648,1576}, {651,692}, {732,3852}, {850,669}, {895,1177}, {912,517}, {916,516}, {952,2818}, {1112,5972}, {1147,156}, {1351,182}, {1370,1619}, {1824,4640}, {1829,960}, {1843,141}, {1897,109}, {1902,9943}, {1986,113}, {1993,184}, {2451,9426}, {2501,8651}, {2574,2574}, {2575,2575}, {2771,2778}, {2850,8674}, {2888,2917}, {2987,1976}, {2996,5023}, {3060,2}, {3146,64}, {3187,31}, {3193,1437}, {3218,10535}, {3219,10536}, {3448,10117}, {3543,10606}, {3564,511}, {3569,5027}, {3575,5907}, {3580,1495}, {3732,101}, {3868,1}, {4240,5502}, {4391,667}, {5095,6593}, {5392,3135}, {5446,140}, {5663,2777}, {5889,4}, {5905,55}, {5921,1350}, {5942,198}, {6152,1209}, {6193,155}, {6241,5878}, {6242,6288}, {6353,8780}, {6368,1510}, {6392,3053}, {6403,1352}, {6515,25}, {7253,3733}, {7576,5891}, {7703,7712}, {7722,7728}, {7754,32}, {7762,39}, {8057,520}, {8673,525}, {8677,900}, {8681,524}, {8878,10329}, {9007,8675}, {9028,674}, {9031,9002}, {9033,526}, {9051,9001}, {9190,9192}, {9308,577}, {9517,690}, {9927,1658}, {9965,2192}, {9979,351}, {10015,1960}, {10340,3499}.

 

Pairs {Z, P}

{1,3868}, {2,3060}, {3,4}, {4,5889}, {5,52}, {6,193}, {25,6515}, {26,68}, {31,3187}, {32,7754}, {39,7762}, {55,5905}, {64,3146}, {101,3732}, {109,1897}, {110,110}, {113,1986}, {140,5446}, {141,1843}, {154,2}, {155,6193}, {156,1147}, {157,317}, {159,69}, {160,264}, {182,1351}, {184,1993}, {198,5942}, {206,6}, {221,145}, {351,9979}, {418,324}, {511,3564}, {512,525}, {513,521}, {516,916}, {517,912}, {520,8057}, {523,520}, {524,8681}, {525,8673}, {526,9033}, {577,9308}, {667,4391}, {669,850}, {674,9028}, {690,9517}, {692,651}, {900,8677}, {924,523}, {960,1829}, {1147,155}, {1154,539}, {1177,895}, {1209,6152}, {1350,5921}, {1352,6403}, {1437,3193}, {1495,3580}, {1498,20}, {1503,511}, {1510,6368}, {1511,113}, {1576,648}, {1619,1370}, {1658,9927}, {1660,394}, {1960,10015}, {1971,385}, {1976,2987}, {2192,9965}, {2328,27}, {2360,29}, {2390,519}, {2393,524}, {2574,2574}, {2575,2575}, {2777,5663}, {2778,2771}, {2781,542}, {2818,952}, {2883,185}, {2917,2888}, {2935,146}, {3053,6392}, {3135,5392}, {3185,92}, {3197,144}, {3357,382}, {3499,10340}, {3556,8}, {3566,512}, {3733,7253}, {3819,428}, {3827,518}, {3852,732}, {4640,1824}, {5023,2996}, {5027,3569}, {5502,4240}, {5878,6241}, {5891,7576}, {5907,3575}, {5972,1112}, {6000,30}, {6001,517}, {6288,6242}, {6593,5095}, {6759,3}, {7712,7703}, {7728,7722}, {8651,2501}, {8674,2850}, {8675,9007}, {8676,514}, {8780,6353}, {9001,9051}, {9002,9031}, {9192,9190}, {9306,25}, {9426,2451}, {9934,74}, {9943,1902}, {10117,3448}, {10192,51}, {10274,195}, {10282,5}, {10329,8878}, {10535,3218}, {10536,3219}, {10537,63}, {10539,24}, {10540,186}, {10606,3543}.