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The Euler pencil of cubics is formed by the isogonal pKs with pivot P = X(i) on the Euler line. Each cubic contains A, B, C, I, Ia, Ib, Ic, O, H. The table at the bottom of this page gives a selection of such cubics. See also the related CL037 and a generalization in Table 54.

 

Points on these cubics

Each cubic with given pivot P defined by OP = k OH (vectors, k real number or infinity) contains the already mentioned points and :

(1) the isogonal conjugate P* of P (P* is the 6th point of the cubic on the Jerabek hyperbola)

(2) the cevian quotient P/M of any point M on the cubic and its isogonal conjugate (P/M)*. Note that P*, M and P/M are three collinear points on the cubic. In particular :

  • P/I, 6th point on the conic through O, H, Ia, Ib, Ic and perspector of IaIbIc and PaPbPc,
  • P/O, 6th point on the Stammler hyperbola (the diagonal conic passing through the in/excenters and O). Note that O, P* and P/O are collinear.
  • P/H, 6th point on the Steiner or Don Wallace hyperbola (the diagonal conic passing through the in/excenters and H). Note that H, P* and P/H are collinear.
  • P/P*. Note that P/O, P/H and (P/P*)* are collinear.

(3) the 6 vertices of the (Kiepert) isosceles triangles erected on the sides of ABC having a base angle 𝝋 such that

cos(2𝝋) = (1 – k) / (2k) <=> k = 1 / (4cos2 𝝋 – 1) = (cot2 𝝋 + 1) / (3 cot2 𝝋 - 1) <=> tan2 𝝋 = 3 – 4 / (k + 1).

These triangles can be drawn externally (triangle AeBeCe) or internally (triangle AiBiCi) and are real if and only if k is not in ]-1;1/3[ i.e. P is not in ]X(20);X(2)[. These six points obviously lie on the perpendicular bisectors of ABC. See Table 32 for other curves related to these triangles. See also rotated cevian lines below.

(4) the perspectors Pi (resp. Pe) of the triangles ABC and AeBeCe (resp. AiBiCi) and their isogonal conjugates Pe*, Pi*. Note that Pe, Pi lie on the Kiepert hyperbola and on a line passing through K and P/O. Hence Pe*, Pi* are the 2nd and 3rd intersections of the cubic with the Brocard line OK.

The tangents to the cubic at Ae, Be, Ce, Pe* (resp. Ai, Bi, Ci, Pi*) concur on the curve.

(5) the vertices Pa, Pb, Pc of the cevian triangle of P.

(6) the points Ha = AH /\ Pa P/O, Hb, Hc (on the altitudes) and their isogonal conjugates Oa = AO /\ Pa P/H, Ob, Oc (on the cevian lines of O).

P/H is the perspector of OaObOc and PaPbPc and P/O is the perspector of HaHbHc and PaPbPc.

table27DarbouxSisters

These points Ha, Hb, Hc are collinear if and only if :

– P = X(20) in which case they are the infinite points of the altitudes,

– P = X(14709) or P = X(14710). These are the common points of the Euler line and the circumconic through X(74) and X(252).

The corresponding points Ha, Hb, Hc lie on the axes of the ellipse K (the inconic with center K, perspector H).

The two cubics pK(X6, X14709) and pK(X6, X14710) have a flex at O and the axes of the ellipse K are the harmonic polars of O in the cubics. The inflexional tangents at O are parallel to the asymptotes of the Jerabek hyperbola.

This has been studied by Fred Lang (Hyacinthos #4382) who called these cubics the Darboux sisters. See figure opposite.

(7) the perspector IO of IaIbIc and OaObOc (the 3rd point of the cubic on the line IH) and its extraversions IOa, IOb, IOc.

(8) the perspector IH of IaIbIc and HaHbHc (the 3rd point of the cubic on the line IO) and its extraversions IHa, IHb, IHc. Thus, IH* is the 6th point of the cubic on the Feuerbach hyperbola and IHa*, IHb*, IHc* are the 6th points of the cubic on the Boutin hyperbolas.

 

Construction of these triangles AeBeCe and AiBiCi with a given pivot P

First, we construct the polar conic C(O) of O in the cubic. C(O) is a rectangular hyperbola passing through O and K, having its asymptotes parallel to those of the Jerabek hyperbola. It also contains the harmonic conjugate of O with respect to H and P.

See below for further properties of C(O).

The perpendicular bisector of BC meets C(O) at O and another point A'. The circle with center the midpoint Ma of BC which is orthogonal to the circle with diameter OA' meets the perpendicular bisector of BC at the requested points Ae and Ai. Similarly we can construct Be and Bi, Ce and Ci. This easily gives all the other mentioned points on the cubic.

 

Rotated cevian lines

With the same notations as in (3) above, let us consider a variable point M on the cubic with pivot P on the Euler line such that OP = 1 / (1 + 2 cos 2 𝝋) OH (vectors). Recall that P lies outside ]X(2), X(20)[.

The cevian line AM is rotated about A with angles +/– 𝝋 producing two lines meeting BC at A1, A2 respectively. The corresponding points B1, B2 on AC and C1, C2 on AB are defined similarly.

One remarkable property is that, for any M on the cubic, these six points always lie on a same conic.

 

Orthic line and poles of the orthic line

table27polesOH

The orthic line of any cubic of the pencil (except K003 since it is a stelloid) is the Euler line. In other words, the polar conic of any point on the Euler line is a rectangular hyperbola.

For a given cubic different of K003, these hyperbolas form a pencil and meet at four points called the poles of the Euler line in the cubic.

The locus of the poles when the pivot traverses the Euler line is a quartic passing through X(3), X(4), X(6), X(1670), X(1671), X(2574), X(2575), X(3413), X(3414).

It has four real asymptotes parallel to those of the Kiepert and Jerabek hyperbolas. It is bitangent at O and H to the Euler line.

The figure represents the Orthocubic K004 and the four corresponding poles, one of them being H. The polar conic of O is the Jerabek hyperbola and that of H is the diagonal hyperbola through H and the in/excenters.

***

Note that the locus of the poles of the line PP* is the cubic K511.

 

Circular polar conics

table27copocer

For any cubic with pivot P of the Euler pencil, there is one and only one point Q whose polar conic is a circle. This point is in a way the Lemoine point of the cubic and we call it the Stuyvaert point of the cubic. See K609 for further details and the table below.

When P traverses the Euler line, the locus of Q is a rectangular hyperbola (H) passing through X(2), X(3), X(6), X(110), X(154), X(354), X(392), X(1201), X(2574), X(2575), X(3167) and the three vertices of the Thomson triangle. These points are the common points (apart A, B, C) of the Thomson cubic K002 and the circumcircle of ABC.

(H) is actually the Jerabek hyperbola for this Thomson triangle. Its center X(5642), labelled X on the figure, is the midpoint of GX(110).

Q coincides with the points X(2), X(3), X(6), X(110) when P is X(3), X(20), X(4), X(30) corresponding to the cubics K003, K004, K006, K001 respectively.

Note that two antipodes Q, Q' on (H) correspond to two points P, P' on the Euler line which are inverse in the circle with center O, radius 3R.

The points Q related with K002 and K005 were added to ETC in June 2014. They are respectively :

X(5544) = a^2*(a^4 - 4*a^2*b^2 + 3*b^4 - 4*a^2*c^2 - 26*b^2*c^2 + 3*c^4) : : , SEARCH = 2.1291814336,

X(5643) = a^2*(a^4 - 3*a^2*b^2 + 2*b^4 - 3*a^2*c^2 - 11*b^2*c^2 + 2*c^4) : : , SEARCH = 1.8976292080.

 

Other polar conics

table27copoX3

Polar conic of O

Recall that the polar conic C(O) of the circumcenter O in every pK of the Euler pencil is a rectangular hyperbola passing through O, K, X(2574), X(2575) hence having its asymptotes parallel to those of the Jerabek hyperbola.

It also contains the harmonic conjugate P' of O with respect to H and P.

All these polar conics are centered on the line X(2), X(98), X(110), X(114), X(125), etc. For example, with P = X(3), X(4), X(631) we obtain the Stammler, Jerabek, Jerabek-Thomson hyperbolas respectively.

See Q002 for other related properties and also K920.

It follows that the two parallels at O to these asymptotes meet the pK again at two pairs of (not always real nor distinct) points {P1, P2), {P3, P4} with same midpoint O.

These four points also lie on the circum-conics which are the isogonal transforms of the parallels to the asymptotes at Q, the P-Ceva conjugate of O. Q obviously lies on the pK.

There are three points on the Euler line such that C(O) is a degenerate rectangular hyperbola.

One of them is X(20) corresponding to the Darboux cubic K004, a central cubic with center O. C(O) splits into the line at infinity and the inflexional tangent at O, namely the Brocard axis.

The two remaining points P1, P2 lie on the circum-conic passing through X(74), X(252).

See "Points on these cubics, (6)" above.

The corresponding cubics pK1, pK2 have an point of inflexion at O with inflexional tangent passing through X(2574), X(2575) respectively.

Remark that the inflexional tangent of pK1 (resp. pK2) meets pK2 (resp. pK1) at two points on the circle with center O passing through H.

C(O) splits into this tangent and its perpendicular at K. Note that the centers Q1, Q2 of these decomposed conics lie on the Brocard circle.

table27copoX3dg
table27copoX4

Polar conic of H

The polar conic of the orthocenter H in every pK of the Euler pencil is a rectangular hyperbola passing through H, the harmonic conjugate of H with respect to O and P, and three other (blue) fixed points which lie :

• on the Jerabek hyperbola (J) corresponding to K003,

• on the diagonal hyperbola (D) passing through the in/excenters of ABC corresponding to K006.

These three points also lie on the circle (C) with center X(9409), orthogonal to the circumcircle (O) and to the Brocard circle. (C) passes through X(15), X(16), X(74), X(112), etc.

The image of (C) under the homothety h(H, 1/2) is the circle (C'), the locus of the centers of all the polar conics of H.

(C') is orthogonal to the circumcircle (O) and to the nine point circle (N). It passes through X(98), X(107), X(125), X(132), X(5000), X(5001), etc, and its center is X(6130).

 

Table of cubics of the Euler pencil

Recall that P is the pivot on the Euler line and Q is the Stuyvaert point (the point with a circular polar conic) on the Jerabek hyperbola (JT) of the Thomson triangle.

Remark 1 : two points P, P' on the Euler line defined by OP = k OH and OP' = k' OH such that 1/k + 1/k' = 2 correspond to two angles 𝝋, 𝝋' with inverse tangents hence 𝝋 + 𝝋' = π/2 i.e. 𝝋' = π/2 - 𝝋. These points are inverse in the circle with diameter OH (abbreviated OH-inverses) or, equivalently, harmonic conjugates with respect to O and H. Some of these pairs of points are highlighted in two cells of same colour in the colum P of the table below. Note that the corresponding points Q, Q' are collinear with X(373).

See X384 and X5999 for example where 𝝋 = ω, the Brocard angle.

Remark 2 : two points P, P' on the Euler line, inverse in the circle C(O, 3R), correspond to two points Q, Q' antipodes on (JT).

***

The cells highlighted in blue in the last column correspond to complex values of 𝝋 since the point P is between X(20) and X(2). They are given for completeness but they have no geometrical signification.

Table built with the cooperation of Peter Moses.

cubic

P = X(i)

Q= X(i)

centers on the cubic : X(1), X(3), X(4) and X(i) for i = ... / remark

𝝋

K002

2

5544

see the page

0

K003

3

2

see the page / the only equilateral cubic of the pencil

 

K006

4

6

see the page

+/- π/4

K005

5

5643

see the page

+/- π/6

K004

20

3

see the page / the only central cubic of the pencil

+/- π/2

 

21

 

21, 65

 

 

22

 

22, 66, 159, 8270

 

 

23

 

23, 67, 2930

 

 

24

 

24, 68, 9937

 

 

25

 

25, 69, 1716, 3186, 3504, 7093

 

 

26

 

26, 70

 

K109

27

 

19, 27, 63, 71, 226, 284, 579, 1751, 1780

 

 

28

 

28, 72, 1724

 

 

29

 

29, 73

 

K001

30

110

see the page / the only circular cubic of the pencil

+/- π/3

 

140

 

140, 1173, 3337, 7161

+/- 1/2 arccos(3/2)

 

186

 

186, 265, 2931

 

 

237

 

237, 290, 3511

 

 

297

 

248, 297

 

K243

376

5646

376, 3426, 5119, 7284

+/- 1/2 arccos(-2)

 

381

5645

381, 3431

+/- arctan(√(3/5))

 

382

9716

382

+/- arctan(√(5/3))

K020

384

 

see the page / contains the vertices of the 1st and 3rd Brocard triangle

+/- ω

 

401

 

401, 1987

 

 

404

 

8, 56, 404, 3216

 

 

407

 

407, 2647, 9398

 

 

408

 

408, 2662

 

 

412

 

412, 1715

 

 

427

 

427, 1176

 

 

429

 

429, 1798

 

 

442

 

442, 1175

 

 

450

 

450, 1092, 1093, 1942

 

 

452

 

452, 2213

 

 

468

 

468, 895

 

 

474

 

474, 3361, 4866

 

 

550

5888

550

+/- 1/2 arccos(-3/2)

K832

631

 

631, 3338, 3527, 7162

+/- 1/2 arccos(2)

 

851

 

851, 1758, 2648, 2655, 2656

 

 

858

 

858, 1177

 

 

1005

 

1005, 1242

 

 

1006

 

1006, 1243, 5902

 

 

1009

 

1009, 1244

 

 

1010

 

10, 58, 386, 1010, 1245

 

 

1011

 

1011, 1246

 

 

1012

 

1012, 2093

 

 

1113

 

1113, 2574

 

 

1114

 

1114, 2575

 

 

1325

 

1325, 2948, 10693

 

 

1585

 

493, 1585, 3068, 6413

 

 

1586

 

494, 1586, 3069, 6414

 

 

1656

 

1656

+/- arctan(1/√7)

 

1657

7712

1657

+/- arctan(√7)

 

1816

 

1816, 1896

 

 

1817

 

610, 1172, 1214, 1817, 1903, 2184

 

 

2071

 

2071, 2935

 

 

2409

 

2409, 2435

 

 

2475

 

191, 267, 2475, 6597

 

 

2476

 

1454, 2476

 

 

2675

 

2671, 2672, 2673, 2674, 2675

+/- arctan(3 – √5)

 

3091

5644

3091, 8888

+/- arctan(1/√2)

 

3146

3167

3146, 3532, 5128, 7285

+/- arctan(√2)

 

3522

 

1697, 3522, 7091

 

 

3523

 

3333, 3523, 7160

 

 

3529

154

3529

 

 

3542

 

1079, 3542, 7042

 

 

3559

 

225, 283, 920, 921, 3559

 

 

3651

 

35, 79, 3651

 

 

4184

 

4184, 8049, 8053

 

 

4188

 

1420, 3680, 4188

 

 

4203

 

1403, 4203, 7155

 

 

4220

 

171, 256, 4220

 

 

5056

 

3311, 3312, 3316, 3317, 5056

+/- arctan(1/2)

K156

5059

 

1131, 1132, 1151, 1152, 5059

+/- arctan(2)

 

5067

 

5067, 6419, 6420, 10194, 10195

+/- 1/2 arccos(4/5)

 

5125

 

34, 78, 1714, 5125

 

K422

5999

 

98, 147, 182, 262, 511, 5999

+/- (π/2 – ω)

 

6353

 

6353, 6391, 8854, 8855

 

 

6820

 

393, 394, 6820

 

 

6840

 

2949, 5535, 6326, 6840

 

 

6905

 

36, 80, 6905

 

 

6909

 

104, 517, 6909

 

 

6940

 

5559, 5563, 6940

 

 

6986

 

942, 943, 6986

 

 

7411

 

7, 55, 7411

 

 

7437

 

885, 2283, 7437

 

 

7471

 

110, 523, 7471

 

 

7488

 

2917, 6145, 7488

 

 

7513

 

580, 581, 7513

 

 

7541

 

1807, 1870, 7541

 

 

7580

 

165, 2947, 3062, 7580

 

 

8613

 

216, 275, 577, 2052, 8612, 8613

 

K751

9855

 

187, 574, 598, 671, 8591

tan 𝝋 = 3 tan ω

 

10486

 

575, 576, 7607, 7608, 7609, 7616, 8914, 10486

cot 𝝋 = 3 tan ω

K152

P152

 

P152 = X(10996), see below

 

 

Z1

 

1691, 1916, 3094, 3407, 8782, Z1

+/- 2ω

 

Z2

 

3095, 3398, 3399, 3406

+/- (π/2 – 2ω)

 

Z3

 

1343, 1671, 1676, 5404

+/- ω/2

 

Z4

 

1342, 1670, 1677, 5403

+/- (π/2 – ω/2)

 

5067'

 

1327, 1328, 6200, 6396

+/- 1/2 arccos(–4/5)

Notes :

P152 = (a^2-b^2-c^2) (a^8-2 a^4 b^4+b^8+20 a^4 b^2 c^2-4 b^6 c^2-2 a^4 c^4+6 b^4 c^4-4 b^2 c^6+c^8): : , on lines {2,3}, {64,141}, {69,185}, {216,7738}, {388,1040}, {497,1038}. P152 and X7386 are OH-inverses.

For this point, we have cos(2𝝋) = cosA cosB cosC / 4.

P152 is now X(10996) in ETC (2016-11-20).

***

Z1 = midpoint of X7924, X9855, on lines {2, 3}, {99, 736}, {187, 5152}, {316, 5149}, {325, 8290}, {385, 698}, {511, 4027}, {1691,1916}, {3094,3407}, {3095,10131}, {3314,4048}, {3329,5116}, {5017,7766}, {5104,8289}, {7761,10000}.

Z2 on lines {2,3}, {99,8295}, {3095,3406}, {3398,3399}, {9983,10104}.

Z3 on lines {2,3}, {6,2546}, {76,1671}, {83,1343}, {2547,5085}, {6248,8161}.

Z4 on lines {2,3}, {6,2547}, {76,1670}, {83,1342}, {2546,5085}, {6248,8160}.

{Z1, Z2} and {Z3, Z4} are pairs of OH-inverses. They are related with Table 38.

Z1, Z2, Z3, Z4 are now X(10997), X(10998), X(10999), X(11000) in ETC (2016-11-20).

***

X5067' is the OH-inverse of X5067, on lines {2,3}, {40,4669}, {388,4324}, {485,6486}, {486,6487}, {497,4316}, {515,4677}, {516,7967}, etc.

X5067' is now X(11001) in ETC (2016-11-21).

***

Additional angles :

X3146 : +/- 1/2 arccos(-1/3).

X9855 : +/- 1/2 arccos[-(4 - 5 cos 2ω) / (5 - 4 cos 2ω)] and X10486 : +/- 1/2 arccos[(4 - 5 cos 2ω) / (5 - 4 cos 2ω)], these two points are OH-inverses.