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Let P = p : q : r be a fixed point and M a variable point. Denote by AmBmCm the pedal triangle of M and by MaMbMc the triangle formed by the orthogonal projections of M on the cevian lines of P.

If P = H, these two triangles are always perspective and the perspector is f(P), equivalently the midpoint of the points M, H/M (H-Ceva conjugate of M) or the barycentric product of M and its orthocorrespondent.

If P is distinct of H, the triangles are perspective if and only if M lies on a circular pK with pivot P whose pole is g(P). The first barycentric coordinate of g(P) is : a^2[-2 SA p^2 + a^2(qr+rp+pq) - p(b^2q+c^2r)].

g(P) is the pole of the isoconjugation which swaps P and igP, the inverse (in the circumcircle) of the isogonal conjugate of P.

 

Circular isogonal pKs

g maps any real point at infinity to the Lemoine point K hence there are infinitely many isogonal circular pKs. The most famous is the Neuberg cubic K001 with P = X(30) = infinite point of the Euler line. See examples in the table below and also Special Isocubics §4.1.1.

P

centers on the cubic

cubic

X30

X1, X3, X4, X13, X14, X15, X16, X30, X74, X370, X399, X484, X616, X617, X1138, X1157, X1263, X1276, X1277, X1337, X1338, X2132, X2133, etc

K001

X512

X1, X99, X512, X2142, X2143, Brocard points

K021

X513

X1, X100, X513, X1054

K661

X515

X1, X36, X40, X80, X84, X102, X515

K269

X518

X1, X105, X518, X1282

 

X522

X1, X109, X522, X1768

 

X524

X1, X2, X6, X111, X524

 

X527

X1, X9, X57, X527, X2291

 

X698

X1, X194, X698, X699

 

X736

X1, X32, X76, X736, X737

 

X744

X1, X31, X75, X744, X745

 

X1503

X1, X20, X64, X147, X1297, X1503

K270

 

Circular pKs with pivot H

The reciprocal image of H under g is the orthic axis. Hence, any circular pK with pole on the orthic axis must have its pivot at H and then, the singular focus is the antipode of the red point on the nine-point circle. The following table gives examples of such cubics. See also CL019.

Pole

centers on the cubic

cubic

X230

X4, X487, X488

 

X232

X4, X114, X371, X372, X511, X2009, X2010

K337

X468

X2, X4, X126, X193, X468, X524, X671, X2374

K209

X523

X4, X127, X525, X1289

 

X647

X4, X122, X520, X1301

 

X650

X4, X123, X521

 

X1886

X4, X118, X516, X917

 

X1990

X4, X13, X14, X30, X113, X1300

K059 = O(X4)

X2489

X4, X112, X115, X512

 

X2501

X4, X107, X125, X523

 

X3003

X3, X4, X131, X155, X265, X403, X1299, X1986

K339

 

Circular pKs with pole W ≠ K and pivot P ≠ H

For any pole W = p : q : r different of K, there is one and only one circular pK with pole W. Its pivot is P = h(W) where h is the reciprocal transformation of g. The first barycentric coordinate of P is : b^2c^2p(q+r-p) - (b^4pr+c^4rp-a^4qr). See Special Isocubics §4.2.1.

Recall that, for any pivot P = u : v : w different of H, there is one and only one circular pK with pivot P. See Table 12 for a selection of such cubics with given pivot P.

Any cubic of this type is actually an isogonal pK with respect to a triangle A2B2C2 where A2 is the intersection of the lines A, gcP and Pa, P/inf where gcP is the isogonal conjugate of the complement of P (a point on the circum-conic with perspector W and on the cubic), PaPbPc is the cevian triangle of P, P/inf is the P-Ceva conjugate of the real infinite point of the cubic. Obviously, inf is the pivot of the cubic when the reference triangle is A2B2C2.

See the Droussent cubic K008 for an example of such triangle A2B2C2.

 

Circular pKs passing through a given point

A circular pK with pivot P contains

the incenter X1 if and only if P lies on the line at infinity (isogonal pKs, see above) or on the Feuerbach hyperbola. In this latter case, the pole lies on K1050 = nK(X32, X513, X6) = cK(#X6, X513).

the centroid X2 if and only if P lies on the Droussent cubic K008. The pole lies on K043.

the circumcenter X3 if and only if P lies on the circumcircle (see inversible pKs) or on the Euler line. In this latter case, the pole W must lie on the Brocard axis OK and the isoconjugate P* of the pivot is the inverse of the isogonal conjugate of P, a point on the Jerabek strophoid K039. The construction is easy since we know P and P* (see Special Isocubics §1.4.3 and §4.2.1).

The inversive image of this cubic in the circumcircle is another cubic of the same type with pivot the inverse of P in the circumcircle and pole the conjugate (on the line OK) of W in the circum-conic with perspector X(184).

For example, with P = X(30) we obtain K001 and K073, with P = X(2) we find K043 and K108.

the orthocenter X4 if and only if P lies on the Ehrmann strophoid K025. More precisely, if P is H, the pole lies on the line at infinity (see CL019). If P is not H, the pole lies on the circum-conic passing through G and K. For example, with P = G, we obtain the Droussent cubic K008.

the Lemoine point X6 if and only if P lies on K273 = pK(X111, X671). The isopivot lies on K108 and the pole lies on K1048 = pK(X32 x X111, X111), the barycentric product X(111) x K043.

the Fermat points X13, X14 if and only if P lies on Kn = K060. The pole lies on K095.

the isodynamic points X15, X16 if and only if P lies on the Neuberg cubic K001. The isopivot lies on K073 and the pole lies on K1049 = pK(X6 x X50, X6), the barycentric product X(6) x K856.

***

More generally, a circular pK with pivot P passes through a given point Q if and only if P lies on a circular circum-cubic K_Q with equation

a^2 v w (w y - v z) [-a^2 (x+y) (x+z) + b^2 x (x+y) + c^2 x (x+z)] = 0

which contains H, Q, agQ, giQ, the infinite point ∞Q of Q-gQ, the Ceva point CQ of Q and iQ (on the circumcircle) and also the points S, T, X, Y such that:

  • S = H-giQ /\ Q-CQ,
  • T = Q-giQ /\ agQ-CQ,
  • X second intersection of the line Q-gQ with the rectangular circum-hyperbola passing through gQ and giQ, isogonal transform of the line OQ,
  • Y second intersection of the line H-∞Q with the rectangular circum-hyperbola passing through CQ.

The pole also lies on a circum-cubic Ω_Q passing through K, Q x igQ and more generally the poles of all the pKs with pivots given above.

Furthermore,

  • K_Q has a flex at Q if and only if Q lies on K028, the Musselman (third) cubic.
  • the polar conics of X(5) and ∞Q in K_Q are always rectangular hyperbolas. In other words, the orthic line of K_Q (and K_gQ) is the line X(5)-∞Q.
  • thus, the polar conic of Q in K_Q is a rectangular hyperbola if and only if Q lies on K005, the Napoleon cubic. In this case, K_Q is a focal cubic. For example, with Q = X(4), X(54) we find K025, K466 respectively.
  • K_Q is a K0 if and only if Q lies on K018, the Brocard (second) cubic, and in this case, it is a pK. For example, with Q = X(111), X(524) we find K273, K008 respectively.

***

When Q lies on the circumcircle, K_Q is psK(Q, tgQ, H) therefore it is tangent at A, B, C to the symmedians. Hence, all these cubics form a pencil and the singular focus lies on the circle C(O, 2R). K_Q meets the sidelines of ABC at the vertices of the cevian triangle of tgQ, the isotomic conjugate of the isogonal conjugate of Q, a point on the Steiner ellipse. The tangent at Q passes through O. The infinite real point of K_Q is the isogonal conjugate of Q and the real asymptote envelopes a deltoid, the homothetic of the Steiner deltoid under h(G,4). See the figure and the table below.

Q

points on the cubic K_Q

cubic

remark

X(74)

X(3), X(4), X(30), X(74)

K447

 

X(98)

X(4), X(76), X(98), X(511)

 

 

X(104)

X(4), X(8), X(104), X(517)

 

 

X(106)

X(1), X(4), X(106), X(519), X(1168), X(1320)

 

 

X(107)

X(4), X(107), X(250), X(520)

 

 

X(108)

X(4), X(59), X(108), X(521)

 

 

X(111)

X(2), X(4), X(6), X(23), X(111), X(524), X(671), X(895)

K273

pK

X(112)

X(4), X(112), X(249), X(525)

 

 

X(733)

X(4), X(83), X(732), X(733), X(1916)

 

 

X(759)

X(4), X(21), X(58), X(80), X(758), X(759), X(1325)

K306

 

X(1141)

X(4), X(54), X(265), X(1141), X(1154)

 

 

X(2291)

X(4), X(9), X(57), X(527), X(1156), X(2291)

 

 

X(2373)

X(4), X(69), X(316), X(1177), X(2373), X(2393)

 

 

X(2380)

X(4), X(14), X(15), X(532), X(1337), X(2380)

 

 

X(2381)

X(4), X(13), X(16), X(533), X(1338), X(2381)

 

 

***

When Q lies on the line at infinity, K_Q contains the vertices Ga, Gb, Gc of the antimedial triangle. Hence, all these cubics form a pencil and the singular focus lies on the circle C(H, R).

Q

points on the cubic K_Q

cubic

remark

X(30)

X(4), X(20), X(30), X(146), X(265), X(1138), X(1294)

K449

 

X(511)

X(4), X(147), X(315), X(511), X(1297), X(1670), X(1671), X(1916)

K1131

 

X(514)

X(4), X(150), X(514), X(1305)

 

 

X(517)

X(4), X(153), X(517), X(1295), X(1320)

 

 

X(519)

X(4), X(8), X(80), X(519), X(2370)

 

 

X(523)

X(4), X(523), X(925)

 

focus X(265)

X(524)

X(2), X(4), X(67), X(69), X(316), X(524), X(671), X(858), X(2373)

K008

pK

X(527)

X(4), X(7), X(329), X(527)

 

 

X(532)

X(4), X(13), X(532), X(616), X(622)

 

 

X(533)

X(4), X(14), X(533), X(617), X(621)

 

 

X(1154)

X(3), X(4), X(1154), X(3153)

 

 

X(2393)

X(4), X(23), X(66), X(895), X(1370), X(2393)

 

 

***

K_Q and K_gQ generate a pencil of circular cubics passing through H and meeting the line at infinity at the same real point S, namely that of the line Q-gQ. This pencil contains :

• pK(H x S, H) where H x S is a point on the orthic axis, see above,

• the decomposed cubic which is the union of the line at infinity and the rectangular circum-hyperbola H_Q passing through the isogonal conjugate of the NK-Transform of Q. Recall that the NK-Transform of Q is the pole of the line Q-gQ in the circum-conic passing through Q and gQ.

It follows that all these cubics must have two other common points. These points are the intersections of the polar of S in H_Q with H_Q.

CL035faisceau

 

Orthic line of a circular pK

A circular pK(Ω, P) always has a proper orthic line (L) since it cannot be a stelloid. Recall that (L) is the locus of points whose polar conic in the pK is a rectangular hyperbola. It is also the Laplacian of the cubic. Note that (L) passes through the real infinite point of the circular pK.

Special cases

• when Ω = X(6), the cubic is an isogonal pK with pivot P on the line at infinity, and then (L) is the line OP. (L) also passes through O when P lies on the Jerabek hyperbola in which case Ω lies on K381 = cK(#X6, X523).

• when P = H = X(4), the cubic is an isogonal pK with respect to the orthic triangle. See CL019. Its pole Ω lies on the orthic axis, its isopivot H* lies on the line at infinity, and then (L) is the line X(5)H*. (L) also passes through X(5) when Ω lies on the circum-conic with perspector X(512) in which case P lies on K025.

***

When P is given, recall that Ω is the barycentric product P x igP and (L) is parallel to the line P, gcP.

(L) passes through a given point Q ≠ O if and only if P lies on a circular circum-cubic K(Q) passing through H and the infinite point of the line OQ. The orthic line of K(Q) is the parallel at X(5) to the line OQ.

Some remarkable cases

• when Q lies on K005, K(Q) is a focal cubic with singular focus on K060.

• when Q is on the line at infinity, K(Q) is the antigonal transform of psK(gQ, tQ, H) with singular focus on the circle C(H, R).

• when Q is on the Euler line, K(Q) passes through X(30), X(265) and its tangent at H passes through X(51) except for Q = X(5) since the cubic is the nodal cubic K025 with node H. The singular focus lies on the line X(4), X(265) and the orthic line is the Euler line itself except K(O) which decomposes into the line at infinity and the Jerabek hyperbola.

• when Q is on the Brocard axis, K(Q) passes through X(511), X(32618), X(32619). The singular focus lies on the Fermat axis and the orthic line is parallel at X(5) to the Brocard axis.

***

The following tables show a good selection of cubics K(Q) obtained with the friendly help of Peter Moses.

Listed cubics

Q

K(Q)

X(i) on K(Q) for i =

1

K529

1, 4, 80, 355, 517, 5903, 14987, 33599

2

K060

4, 5, 13, 14, 30, 79, 80, 265, 621, 622, 1117, 1141, 5627, 6761, 11581, 11582, 11584, 11600, 11601, 14372, 14373, 14451, 14452, 17405, 17406, 19658, 33664, 34295, 34296, 34297, 34298, 34299, 34300, 34301, 34302, 34303, 34304, 34305

4

K530

4, 30, 265, 477, 7728, 14989

5

K025

4, 30, 265, 316, 671, 1263, 1300, 5080, 5134, 5203, 5523, 5962, 10152, 11604, 11605, 11703, 13495, 16172, 19552, 31862, 31863, 34150, 34169, 34170, 34171, 34172, 34173, 34174, 34175, 34239, 34240, 37888

6

K337

4, 114, 371, 372, 511, 2009, 2010, 3563, 5000, 5001, 32618, 32619

30

K449

4, 20, 30, 146, 265, 1138, 1294

61

K1132b

4, 14, 15, 62, 511, 621, 1337, 5617, 11601, 32618, 32619, 34219

62

K1132a

4, 13, 16, 61, 511, 622, 1338, 5613, 11600, 32618, 32619, 34220

182

K1134

4, 76, 98, 511, 1342, 1343, 5004, 5005, 9469, 11610, 15407, 32618, 32619, 34237, 34238, 34239, 34240

195

K465

3, 4, 5, 1154, 1157, 1263, 14072, 14979, 14980, 19552, 24772, 33565

381

K427

4, 30, 265, 382, 1539, 20480, 33641

511

K1131

4, 147, 315, 511, 1297, 1670, 1671, 1916, 5002, 5003, 32618, 32619, 34137, 34214

524

K008

2, 4, 67, 69, 316, 524, 671, 858, 2373, 11061, 13574, 14360, 14364, 34163, 34164, 34165, 34166

576

K305

4, 262, 316, 511, 671, 842, 6054, 9970, 32618, 32619, 34235, 34241

 

 

 

Focal cubics

Q

K(Q)

X(i) on K(Q) for i =

1

K529

1, 4, 80, 355, 517, 5903, 14987, 33599

4

K530

4, 30, 265, 477, 7728, 14989

5

K025

4, 30, 265, 316, 671, 1263, 1300, 5080, 5134, 5203, 5523, 5962, 10152, 11604, 11605, 11703, 13495, 16172, 19552, 31862, 31863, 34150, 34169, 34170, 34171, 34172, 34173, 34174, 34175, 34239, 34240, 37888

17

 

4, 13, 622, 11600, 11602, 20428

18

 

4, 14, 621, 11601, 11603, 20429

54

 

3, 4, 1141, 1154, 2070, 25043, 25044, 33565

61

K1132b

4, 14, 15, 62, 511, 621, 1337, 5617, 11601, 32618, 32619, 34219

62

K1132a

4, 13, 16, 61, 511, 622, 1338, 5613, 11600, 32618, 32619, 34220

195

K465

3, 4, 5, 1154, 1157, 1263, 14072, 14979, 14980, 19552, 24772, 33565

627

 

4, 13, 17, 622, 633, 3479, 8174, 11600, 34219, 36766

628

 

4, 14, 18, 621, 634, 3480, 8175, 11601, 34220

2120

 

4, 3481, 34304

2121

 

4, 3482

3336

 

4, 79, 484, 517, 5902, 13143

3459

 

4, 1263, 11584, 19552, 24144, 25148, 31392

3460

 

4, 7165, 34300

3461

 

4, 3483

3462

 

4, 6761, 8439

3463

 

4, 3484, 33664

3467

 

4, 3065, 19658

3468

 

4, 3466, 34301

3469

 

4, 3465, 34303

3470

 

4, 399, 5627, 5663, 33567

3471

 

4, 1117, 1138, 3448

3489

 

4, 1337

3490

 

4, 1338

6191

 

4, 7059

6192

 

4, 7060

7344

 

4, 1276, 17405

7345

 

4, 1277, 17406

8837

 

4, 1337, 8175, 8471

8839

 

4, 1338, 8174, 8479

8918

 

4, 8462

8919

 

4, 8452

8929

 

4, 8172, 11581

8930

 

4, 8173, 11582

 

 

 

Unlisted cubics through at least six ETC centers

Q

K(Q)

X(i) on K(Q) for i =

15

 

4, 511, 633, 3389, 3390, 22507, 32618, 32619

16

 

4, 511, 634, 3364, 3365, 22509, 32618, 32619

17

 

4, 13, 622, 11600, 11602, 20428

18

 

4, 14, 621, 11601, 11603, 20429

21

 

1, 4, 30, 265, 3065, 6914

32

 

4, 511, 1687, 1688, 2698, 12177, 32618, 32619

39

 

4, 511, 1352, 1689, 1690, 32618, 32619

54

 

3, 4, 1141, 1154, 2070, 25043, 25044, 33565

58

 

1, 4, 511, 5018, 10446, 32618, 32619

140

 

2, 4, 30, 98, 265, 3448, 12079

283

 

1, 4, 68, 74, 3465, 13754

371

 

4, 511, 637, 3385, 3386, 6231, 32618, 32619

372

 

4, 511, 638, 3371, 3372, 6230, 32618, 32619

386

 

4, 80, 256, 511, 32618, 32619

389

 

4, 186, 317, 511, 1299, 32618, 32619

517

 

4, 153, 517, 1295, 1320, 3436

519

 

4, 8, 80, 519, 2370, 5176, 21290

527

 

4, 7, 329, 527, 3254, 5057

532

 

4, 13, 532, 616, 622, 11122, 11600, 22649

533

 

4, 14, 533, 617, 621, 11121, 11601, 22648

549

 

4, 8, 30, 265, 631, 12317

569

 

4, 311, 511, 1141, 32618, 32619

575

 

2, 4, 23, 111, 263, 511, 11185, 32618, 32619

577

 

4, 511, 1298, 5562, 32618, 32619

578

 

4, 250, 264, 511, 1300, 5962, 15463, 32618, 32619

627

 

4, 13, 17, 622, 633, 3479, 8174, 11600, 34219, 36766

628

 

4, 14, 18, 621, 634, 3480, 8175, 11601, 34220

758

 

1, 4, 758, 3869, 5080, 11604

970

 

4, 386, 511, 573, 11609, 32618, 32619

1125

 

4, 7, 80, 516, 4295, 5816

1154

 

3, 4, 1154, 3153, 18401, 33565

1385

 

4, 8, 104, 517, 12247, 34242

1493

 

3, 4, 1154, 2383, 33565, 37943

1503

 

4, 1503, 12384, 34168, 34239, 34240

1656

 

4, 30, 115, 265, 381, 10113, 13530, 33601

1790

 

4, 68, 74, 1276, 1277, 13754

1994

 

4, 15, 16, 1263, 6243, 16837, 19552

2055

 

4, 511, 577, 578, 3484, 32618, 32619

2092

 

4, 511, 1685, 1686, 32618, 32619

2393

 

4, 23, 66, 895, 1370, 2393

2888

 

4, 1141, 1263, 6288, 19552, 32423

3094

 

4, 511, 3102, 3103, 32618, 32619

3095

 

4, 39, 511, 5207, 32618, 32619, 37841

3098

 

4, 511, 7768, 29011, 32618, 32619

3336

 

4, 79, 484, 517, 5902, 13143

3357

 

4, 2693, 6000, 10152, 15318, 34170

3368

 

4, 511, 3379, 3380, 32618, 32619

3379

 

4, 511, 3395, 3396, 32618, 32619

3393

 

4, 511, 3368, 3369, 32618, 32619

3395

 

4, 511, 3393, 3394, 32618, 32619

3398

 

4, 32, 182, 511, 18906, 32618, 32619

3459

 

4, 1263, 11584, 19552, 24144, 25148, 31392

5012

 

3, 4, 1154, 3438, 3439, 5899, 13597, 33565

5019

 

4, 511, 10441, 13332, 13333, 32618, 32619

5092

 

4, 511, 5984, 29180, 32618, 32619

5398

 

4, 511, 953, 17139, 32618, 32619

5476

 

4, 316, 671, 14492, 19924, 31670, 32271

5694

 

4, 104, 2771, 5080, 5887, 11604

5889

 

3, 4, 1154, 18403, 22751, 33565

5893

 

4, 20, 10152, 11744, 13219, 15077, 34170

6102

 

3, 4, 562, 1154, 32710, 33565

6199

 

4, 511, 1151, 3312, 32618, 32619

6395

 

4, 511, 1152, 3311, 32618, 32619

6425

 

4, 511, 6200, 6420, 32618, 32619

6426

 

4, 511, 6396, 6419, 32618, 32619

6642

 

4, 25, 30, 265, 1112, 3563

6644

 

4, 24, 30, 265, 1299, 1986

6696

 

4, 253, 1503, 10152, 12324, 13573, 34170

6759

 

4, 1294, 6000, 14249, 34239, 34240

7617

 

4, 316, 671, 17503, 32479, 33601

7751

 

4, 76, 316, 538, 671, 2367

7758

 

4, 67, 69, 141, 524, 5207, 33665

7759

 

4, 83, 315, 316, 671, 754, 6328

7764

 

2, 4, 754, 2857, 11606, 20022, 36163, 37841

8542

 

4, 66, 111, 316, 671, 895, 2393, 10422, 11188

8546

 

4, 23, 67, 69, 524, 5354, 10630

8548

 

4, 25, 111, 858, 2987, 14984

8550

 

4, 67, 69, 98, 524, 5485, 7735, 16092

9019

 

4, 6, 22, 5189, 9019, 18125

9690

 

4, 511, 6409, 6418, 32618, 32619

9737

 

4, 511, 2710, 9289, 32618, 32619

10282

 

4, 20, 23, 1297, 6000, 32319

10627

 

3, 4, 1154, 5189, 16030, 29011, 33565

11126

 

3, 4, 13, 15, 622, 1154, 5616, 11600, 33565, 36981

11127

 

3, 4, 14, 16, 621, 1154, 5612, 11601, 33565, 36979

11165

 

4, 67, 69, 524, 2482, 15069, 15534, 33601

11178

 

4, 316, 671, 3818, 11645, 14458

11271

 

4, 1263, 1487, 3519, 5900, 19552

11438

 

4, 340, 511, 32618, 32619, 32710

12038

 

4, 20, 68, 74, 186, 13754, 15469

12161

 

3, 4, 403, 1154, 1299, 8883, 14111, 33565

13323

 

4, 58, 104, 314, 511, 572, 32618, 32619

13335

 

4, 511, 6776, 9292, 32618, 32619

13336

 

4, 511, 1232, 13597, 32618, 32619

13346

 

4, 511, 1294, 14615, 32618, 32619

13352

 

4, 477, 511, 3260, 32618, 32619

14520

 

4, 511, 991, 4253, 32618, 32619

14627

 

4, 511, 1263, 3447, 3613, 10263, 19552, 32618, 32619

15311

 

4, 3346, 6225, 10152, 15311, 34170

15345

 

3, 4, 195, 1154, 2888, 11584, 33565

15733

 

4, 1156, 3434, 6601, 15733, 17615

15774

 

4, 30, 265, 340, 2133, 5667

15781

 

4, 30, 265, 3484, 8431, 21659

16266

 

3, 4, 97, 858, 1154, 1297, 33565

17733

 

4, 80, 314, 740, 4647, 20558

22802

 

4, 1294, 2777, 5878, 10152, 34170

23329

 

4, 10152, 14216, 15319, 18400, 34170

32046

 

3, 4, 23, 98, 1154, 11140, 14652, 33565

34117

 

4, 1297, 2781, 5523, 11605, 19149

34507

 

4, 98, 316, 542, 671, 1352

34508

 

4, 14, 316, 531, 621, 671, 11601

34509

 

4, 13, 316, 530, 622, 671, 11600

35195

 

3, 4, 484, 1154, 3065, 33565

37814

 

4, 30, 186, 265, 7722, 32710