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A more complete description can be found in the FG paper "Orthocorrespondence and orthopivotal cubics". See the Downloads page. Here is a short summary with several additional informations. See also the related Droz-Farny cubics in CL039 and also Table61.

First recall that the orthocorrespondent of P is the point, here denoted by oc(P), defined as follows : the perpendiculars drawn through P to the lines AP, BP, CP meet BC, CA, AB respectively at three collinear points called the orthotraces of P. These three points lie on a line called orthotransversal of P. The trilinear pole of this line is oc(P). Now, in general, there are two points P1 and P2 sharing the same orthocorrespondent P. These points (although not always real) are said to be orthoassociates and they are the antiorthocorrespondents of P.

An orthopivotal cubic is denoted by O(P) where P is its orthopivot. It is equivalently :

  • the locus of M such that the points P, M, oc(M) are collinear.
  • the locus of the intersections of a line through P with the circle which is its antiorthocorrespondent
  • the locus of the perspectors of the equilateral triangles centered at P which are perspective with ABC. (Jean-Pierre Ehrmann)

O(P) is a circular circum-cubic and passes through the Fermat points X(13) and X(14). It is a K0 for every point P.

With P = (u : v : w), the equation of O(P) is :

x [(c^2 u – 2 SB w) y^2 – (b^2 u – 2 SC v) z^2]= 0.

 

Denote by F1, F2 the foci of the Steiner inellipse. The singular focus F of O(P) is Psi(P), the Psi-transform of P. Psi(P) is the reflection in the line F1F2 of the inverse of P in the circle with diameter F1F2. Psi is clearly an involution hence F = Psi(P) <=> P = Psi(F).

F is also the center of the polar conic of P (a rectangular hyperbola) in the cubics K003 (McCay) and K024 (Kjp). See the related Psi-cubics in Table 60.

Here is a list of pairs {P, Psi(P)} by Peter Moses (updated 2017-01-05).

{1,1054}, {3,110}, {4,125}, {5,3448}, {6,111}, {13,14}, {15,16}, {20,5972}, {23,182}, {25,5622}, {36,5197}, {39,9998}, {69,126}, {98,1316}, {99,5108}, {100,1083}, {105,5091}, {115,6792}, {147,11007}, {184,186}, {187,353}, {193,6719}, {352,574}, {371,7599}, {372,7598}, {376,5642}, {381,9140}, {403,1899}, {468,6776}, {549,9143}, {599,10717}, {616,619}, {617,618}, {621,624}, {622,623}, {671,9169}, {691,9129}, {846,5529}, {858,1352}, {868,11005}, {1151,7602}, {1152,7601}, {1302,6795}, {1340,5639}, {1341,5638}, {1379,6141}, {1380,6142}, {1637,6794}, {1992,9172}, {2070,5012}, {2071,9306}, {2072,11442}, {2549,5913}, {2715,8429}, {3098,7711}, {3111,9147}, {3120,6788}, {3146,6723}, {3165,3166}, {3642,5979}, {3643,5978}, {3734,5971}, {3821,5211}, {3923,5205}, {4427,6789}, {5026,6031}, {5077,9759}, {5085,9157}, {5092,9999}, {5112,9744}, {5118,9153}, {5159,5921}, {5463,5464}, {5465,5466}, {5651,7464}, {5652,9828}, {5653,5968}, {5655,9159}, {6322,9100}, {6787,9148}, {7426,11179}, {7575,11003}, {7998,8724}, {8371,9144}, {10989,11178}, {11162,11184}.

***

Some special cases of O(P) are listed below :

• O(P) is a pK if and only if P lies on the Napoleon cubic K005 and, in this case, the pivot of O(P) lies on the cubic Kn = K060 = O(X5) and its pole on the cubic Co = K095. The isoconjugate of the pivot lies on the Neuberg cubic K001. The singular focus F lies on Q041. See the blue cells in the table.

• O(P) is a nK (in fact a nK0) if and only if P lies on the cubic K397. F lies on a complicated bicircular quintic.

• O(P) is a focal if and only if P lies on the Brocard (second) cubic K018. In this case, F also lies on K018 which is therefore invariant under Psi. See Table 60 for other Psi-invariant cubics.

• O(P) is a K0+ if and only if P lies at infinity (in which case F = G) or on an axis of the Steiner ellipse (in which case F also lies on this same axis).

• O(P) is singular if and only if P lies on Q015 giving either decomposed cubics or nodal cubics. See the orange cells in the table.

The table gives a selection of interesting O(P) with singular focus F and its isogonal transform O*(P).

P

O(P)

O*(P)

F

remarks about O(P)

X(1)

K058

K206

X(1054)

pK

X(2)

line at infinity and Kiepert hyp.

(O) and Brocard axis

 

decomposed cubic

X(3)

K001 Neuberg

K001 Neuberg

X(110)

isogonal pK

X(4)

K059

K114

X(125)

pK

X(5)

K060 Kn

K073

X(3448)

pK

X(6)

K018 Brocard 2

K018 Brocard 2

X(111)

isogonal focal nK0

X(11)

7, 11, 13, 14, 80, 528

15, 16, 36, 55, 59, 840

 

nodal cubic

X(13)

K061a

15, 16, 2378

X(14)

strophoid

X(14)

K061b

15, 16, 2379

X(13)

strophoid

X(15)

K262a

K262b

X(16)

focal

X(16)

K262b

K262a

X(15)

focal

X(20)

K313

K523

X(5972)

 

X(23)

4, 13, 14, 23, 30, 842

K524

X(182)

 

X(32)

K291

K290

?

 

X(39)

K290

K291

X(9998)

 

X(51)

K062

K468

?

F = X3X148 /\ X4X32 /\ X5X147

X(54)

K112

K050

?

inversible pK

X(61)

K261b

K261a

?

pK

X(62)

K261a

K261b

?

pK

X(69)

K442

6, 15, 16, 25, 11, 2393

X(126)

 

X(110)

(O) and Fermat axis

line at infinity and conic

 

decomposed cubic

X(111)

K063

K435

X(6)

focal

X(182)

K292

K263

X(23)

 

X(195)

K067

K439

?

pK

X(511)

K263

K292

X(2)

K0+

X(523)

K064

K148

X(2)

nK0+

X(524)

K065

K640

X(2)

central focal nK0++

X(627)

K066a

15, 16, 61, 1327, 2380

?

pK

X(628)

K066b

15, 16, 62, 1328, 2381

?

pK

X(858)

K479

3, 15, 16, 23, 74, 378, 1177, 2781

X(1352)

 

X(1316)

K023

3, 15, 16, 74, Brocard points

X(98)

the only O(P) through the Brocard points

X(1992)

K452

6, 15, 16, 111, 895

X(9172)

 

F1

K057a

 

F1

two central cubics

F1, F2 are the real foci of the Steiner inellipse

F2

K057b

 

F2

A, B, C

K053-A-B-C

Apollonius circles

 

strophoids

X(3413)

K293a

 

X(2)

axial cubic

X(3414)

K293b

 

X(2)

axial cubic

P450

K450

 

X(868)

axial cubic

P451

K451

 

X(1649)

axial cubic

X(376)

K808

3, 15, 16, 74

X(5642)

 

X(30)

4, 13, 14, 30, 477

3, 15, 16, 74, 5663

X(2)

 

X(57)

1, 13, 14, 57, 226, 484, 537, 1785

1, 9, 15, 16, 284, 1725, 2291, 3063

?

 

X(542)

13, 14, 98, 542

K641

X(2)

 

X(1340)

13, 14, 15, 16, 1340

13, 14, 15, 16, 1380

X(5639)

K0+

X(1341)

13, 14, 15, 16, 1341

13, 14, 15, 16, 1379

X(5638)

K0+

X(1993)

2, 3, 13, 14, 403, 524, 1157, 1993, 3563

K441

?

 

X(1994)

2, 5, 13, 14, 186, 249, 524, 1994

6, 15, 16, 64, 111, 115, 265, 2963

?

 

X(2979)

3, 13, 14, 262, 511, 1157, 1297, 2979

4, 15, 16, 98, 182, 1263, 1503, 2980

?

 

X(5012)

3, 13, 14, 98, 542, 1157, 1687, 1688

K440

?

 

X(5967)

13, 14, 98, 542, 5967

15, 16, 511, 842, 5968

?

nK0

X(5968)

6, 13, 14, 111, 523, 2395, 5466, 5968

2, 15, 16, 110, 524, 2421, 5467, 5967

X(5653)

nK0

X(11422)

3, 6 ,13, 14, 111, 187, 1157, 11422

K304

?

 

 

 

 

 

 

Notes

• X(3413) and X(3414) are the points at infinity of the asymptotes of the Kiepert hyperbola and also those of the axes of the Steiner ellipses.

• P450 is the reflection of X(98) in the perpendicular bisector of the Fermat points. SEARCH = -3.1454529314. P450 is now X(11005) in ETC (2016-11-23).

• P451 is the intersection of the parallel at X(99) and the perpendicular at G to the Fermat line. SEARCH = 5.1605352720. P451 is now X(11006) in ETC (2016-11-23).

***

Special pencils of orthopivotal cubics

• when P lies on the line at infinity, O(P) has its singular focus at G. It is the locus of M such that the Euler line of the antipedal triangle of M passes through P. On the other hand, O*(P) has its singular focus at X(23). It is the locus of M such that the Euler line of the pedal triangle of M passes through P. See the green lines in the table above and also Q002, Q003 for related quartics.

• when P lies on the Euler line, O(P) contains X(4), X(30) and the singular focus lies on the line X(2), X(98), X(110), etc.

• when P lies on the Brocard axis, O(P) contains X(15), X(16) and the singular focus lies on the Parry circle. It follows that O(P) and O*(P) belong to a same pencil of circular circum-cubics passing through X(13), X(14), X(15), X(16) generated by K001 and K018. See CL034.

• when P lies on the line X(3), X(54), X(97), etc, O(P) contains X(3), X(1157) and the singular focus lies on the circle passing through X(2), X(110), X(2070), etc, which is the Psi-image of the line. These are the isogonal transforms of the cubics of the Neuberg pencil. See also Neuberg cubics.

• when P lies on the line X(2), X(98), X(110), X(114), X(125), etc, O(P) contains X(98), X(542) and the singular focus lies on the Euler line. O*(P) is a K0+ passing through X(15), X(16), X(511), X(842) with singular focus also on the Euler line since it is the inverse in the circumcircle of the focus of O(P).