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X(5000) and X(5001) are the antiorthocorrespondents of the Lemoine point X(6). These points are real if and only if ABC is acutangle.

Their midpoint is X(468) and their barycentric product is X(232). Their perpendicular bisector is the orthic axis.

They lie

• on the Euler line,

• on the circumconic with perspector X(3569) passing through X6, X232, X250, X262, X264, X325, X511, X523, X842, X1485, X2065,

• on the diagonal conic passing through X6, X157, X230, X523, X1503.

• on the circles which are the antiorthocorrespondents of the lines passing through X(6). These circles are centrered on the orthic axis and are orthogonal to the circumcircle and more generally to all the circles mentioned below. Note that a line meets the associated circle on K018 since K018 is the orthopivotal cubic O(X6).

They are inverse in any circle of the coaxal system containing the circumcircle, the nine-point circle, the orthocentroidal circle, the orthoptic circle of the Steiner inellipse, the polar circle (hence they are orthoassociate points), the tangential circle. Recall that the radical axis of this system is the orthic axis.

The following table shows a selection of listed curves passing through these two points (which are not repeated in the table). Some of these curves are generalized below.

curve

type

X(i) on the curve for i =

notes

K018

isogonal focal nK0

2, 6, 13, 14, 15, 16, 111, 368, 524

notes 1, 7, 8

K270

isogonal circular pK

1, 20, 64, 147, 1297, 1503, 5018, 7281

note 5

K336

inversible circular pK

3, 98, 485, 486, 1687, 1688, 2065, 3564

notes 2, 5

K337

antigonal circular pK

4, 114, 371, 372, 511, 2009, 2010, 3563

notes 3, 5

K570

circular psK

3, 32, 98, 132, 511, 5403, 5404, 5976, 9467

note 5

K608

central nK0

468, 523

note 4

K828

circular psK

384, 698, 733, 1916, 2076, 3224

note 5

K829

circular nK

112, 2799, 4235

note 7

K1091

focal cubic

6, 523, 1316, 3569

note 12

K1092

focal cubic

6, 30, 1379, 1380, 11472, 13872, 16303

 

 

Q019

circular quartic

2, 3, 187, 3413, 3414

note 8

Q021

bicircular isogonal sextic

1, 3, 4, 1113, 1114, 2574, 2575

note 10

Q024

bicircular sextic

3, 5, 1113, 1114

note 10

Q026

quartic

1, 2, 3, 241, 3229

note 11

Q037

inversible bicircular quintic

1, 3, 15, 16, 30, 36

note 9

Q049

quartic

3, 15, 16, 23, 111, 187, 6104, 6105

 

Q054

bicircular quintic

4, 5, 15, 16, 30

note 9

Q098

inversible circular quartic

3, 6, 187, 2574, 2575, 3513, 3514

note 11

Q115

circular quartic

1, 2, 20, 485, 486, 1323, 3413, 3414

note 8

Q116

circular quartic

2, 17, 18, 550, 3413, 3414

note 8

Q117

circular quartic

2, 4, 3413, 3414

note 8

Q118

circular quartic

2, 376, 3413, 3414

note 8

Q144

quartic

2, 6, 3231

note 11

Q146

circular quartic

3, 4, 69, 2574, 2575, 5167

note 11

Q147

quartic

3, 523, 21444

note 11

Note 1 : more generally, any nK0(Ω, X523) with pole Ω on the line X6, X25 contains X5000, X5001 and also X6. In particular nK0(X2393, X523) is a K+. K018 is the only circular cubic of this pencil whose inverse in the circumcircle is the quartic Q049.

Note 2 : more generally, any pK(Ω, X98) with pole Ω on the line X4, X32 contains X5000, X5001 and obviously X98. K336 is the only circular cubic of this pencil.

Note 3 : more generally, any pK(X232, P) with pivot P on the Euler line contains X5000, X5001 and the four square roots of X232. Recall that X232 is the barycentric product of these two points. K337 is the only circular cubic of this pencil.

Note 4 : more generally, any nK0(Ω, X6) with pole Ω on the orthic axis contains X5000, X5001 and also X523. K608 is the only K+ (and even K++) of this pencil.

Note 5 : more generally, a circular psK passing through X5000, X5001 must have

• its pseudo-pole on K782 = pK(X2211, X25) through X(i) for i = 4, 6, 25, 232, 237, 248, 511, 694, 3053, 3186, 3563,

• its pseudo-pivot on K780 = pK(X4, X297) through X(i) for i = 2, 4, 69, 98, 230, 297, 393, 694, 1503, 1987, 6330, 9473,

• its pseudo-isopivot on K783 = pK(X237, X3) through X(i) for i = 2, 3, 6, 154, 232, 237, 511, 1297, 1976, 1987, 3164, 9292, 9475.

Remark that some of these psKs are in fact pKs.

Note 6 : the following nK0s also contain X5000, X5001 :

• any nK0(X232, P) with root P on the line X6, X523,

• any nK0(X325 x X2501, P) with root P on the line X6, X264,

• any nK0(X237 x X648, P) with root P on the line X250, X523.

Note 7 : more generally, a circular nK passing through X5000, X5001 must have

• its pole on nK0(X2211, X25) through X(i) for i = 6, 112, 232,

• its root on nK0(X4, X297) through X(i) for i = 2, 4, 523, 648.

Remark that some of these nKs are in fact nK0s.

Note 8 : all these circular quartics are in a same pencil which also contains the union of K018 with the line at infinity.

Note 9 : Q037 and Q054 are both bicircular quintics which generate a pencil of quintics of the same type. One is decomposed into K018 and the line at infinity counted twice.

Note 10 : Q021 and Q024 are both bicircular sextics which generate a pencil of sextics of the same type. One is decomposed into Q019 and the circumcircle.

Note 11 : see K1065 for related topics.

Note 12 : more generally, let P be a finite point not lying on the orthic axis and let Ω be a variable point lying on the orthic axis. The circle with center Ω orthogonal to the circumcircle (O) must contain X(5000) and X(5001). It meets the line PΩ at two points of a focal cubic with singular focus P, passing through X(523), X(5000), X(5001) and the common points of (O) and the orthic axis.