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See also K584 for a trilinear equation and related comments

X(1), X(3), X(4), X(5), X(17), X(18), X(54), X(61), X(62), X(195), X(627), X(628), X(2120), X(2121), X(3336), X(3459), X(3460), X(3461), X(3462), X(3463), X(3467), X(3468), X(3469), X(3470), X(3471), X(3489), X(3490), see the table below

Ia, Ib, Ic excenters

Ha, Hb, Hc projections of G on the altitudes

Oa, Ob, Oc their isogonal conjugates

centers of the 6 equilateral triangles erected on the sides of ABC, externally (Ae, Be, Ce) or internally (Ai, Bi, Ci), see the figure below

centers of equilateral cevian and precevian (anticevian) triangles (Jean-Pierre Ehrmann). See table 10, table 14.

Ix-anticevian points and their isogonal conjugates : see table 23

The Napoleon-Feuerbach cubic is the isogonal pK with pivot X(5) = nine-point center. See Table 27.

It is anharmonically equivalent to the Neuberg cubic. See Table 20.

Locus properties : (see also Z. Cerin's paper in the bibliography)

  1. Denote by PaPbPc the pedal triangle of point P. The Euler lines of APbPc, BPcPa, CPaPb concur if and only if P lies on the Napoleon cubic.
  2. Under the homothety h(P,-2), PaPbPc is transformed into QaQbQc which is perspective to ABC if and only if P lies on the Napoleon cubic. For this reason, the Napoleon cubic is called -2-pedal cubic in Pinkernell's paper. The perspector lies on the Kn cubic which is the 2-cevian cubic.
  3. Pa, Pb, Pc are the reflections of point P in the sidelines of triangle ABC. Oa, Ob, Oc are the circumcenters of triangles PaBC, PbCA, PcAB. The triangles ABC and OaObOc are perspective if and only if P lies on Kn (together with the circumcircle and the line at infinity). The locus of the perspector is the Napoleon cubic.
  4. The cevian lines of point P meet the perpendicular bisectors of ABC at A', B', C'. The locus of P such that the sum of oriented lines angles (BA',BC)+(CB',CA)+(AC',AB) = pi/2 (mod. pi) is the Napoleon cubic. When this sum is 0 (mod. pi), we obtain the Brocard (second) cubic K018. (Jean-Pierre Ehrmann)
  5. Let Q be a fixed point, QaQbQc its pedal triangle and P a variable point. The parallel to QbQc through P intersects AB and AC at Ab and Ac. A1 (resp. A2) is the apex of the equilateral triangle erected outwardly (resp. inwardly) on AbAc. Similarly define B1, B2 and C1, C2. The two triangles ABC, A1B1C1 (resp. A2B2C2) are perspective for all P if and only if Q lies on the Napoleon cubic. The perspector lie on K129b (resp. K129a). (Problem raised by Antreas Hatzipolakis, Hyacinthos #10560)
  6. An orthopivotal cubic O(P) is a pK if and only if its orthopivot P lies on the Napoleon cubic. The singular focus of O(P) lies on Q041, the Psi-transform of K005.

Points on the Napoleon cubic

K005hauteurs

Altitudes and cevians of O

K005 contains :

Ha, Hb, Hc projections of G on the altitudes of ABC

– #O = X(3470), the common tangential of O and Ha, Hb, Hc

– Oa, Ob, Oc isogonal conjugates of Ha, Hb, Hc, these points on the cevians of O

– #H, the common tangential of H and Oa, Ob, Oc

Note that the tangents at O and H pass through X(74) and that #O, #H and X(54) are colinear (they are the tangentials of O, H and X(5)) on the satellite line of the Euler line.

– (#O)* = X(3471) and (#H)*, isogonal conjugates of #O and #H

K005triangles

Napoleon triangles

K005 contains :

– the vertices of the outer triangle AeBeCe. The tangents at Ae, Be, Ce, X61 concur at #X61 on the curve.

– the vertices of the inner triangle AiBiCi. The tangents at Ai, Bi, Ci, X62 concur at #X62 on the curve.

– the third points Ae', Be', Ce' on the sidelines of AeBeCe (these two triangles are perspective at X61).

– the third points Ai', Bi', Ci' on the sidelines of AiBiCi (these two triangles are perspective at X62).

Note that #O is also the perspector of the triangles Ae'Be'Ce' and Ai'Bi'Ci' and that #O, #X61, #X62 are colinear on the satellite line of the Brocard line.

K005table10

Centers of the equilateral cevian or anticevian triangles

K005 contains the six centers (not always real as seen in the figure) of the equilateral cevian points i.e. the points (such as X(370)) whose cevian triangle is equilateral. These equilateral cevian points lie on the Neuberg cubic.

These centers lie on the perpendicular bisectors of X(13)X(15) (red points) and X(14)X(16) (blue points).

See Table 10 for more details.

 

K005 also contains the six centers (not always real) of the equilateral anticevian points i.e. the points whose anticevian triangle is equilateral.

See Table 14 for more details and a figure.

 

This gives 24 points on K005, counting the isogonal conjugates.

K005table23

Ix-anticevian points

K005 contains the four (green points, not always real) Ix-anticevian points. These points are the common points (apart A, B, C, H, X5) of the cubics of the pencil which contains K005, K049, K060 and many other cubics.

See Table 23 for explanations.

Obviously K005 also contains the isogonal conjugates (purple points) of these four points.

This gives eight more points on the cubic.

The table gives centers on the cubic. Each weak point (in red) must have its three extraversions on the curve. The SEARCH value is given for these points.

Most of the points are the perspectors of two inscribed triangles in the cubic. The A-vertex is given in the table.

P* is the isogonal conjugate of P and P/Q is the cevian quotient of P and Q.

number

0

1

2

3

4

5

6

7

8

9

10

11

12

triangle

A

Ia

A-cevian of X(5)

Ha

Oa

Ae

Ai

Ae*

Ai*

Ae'

Ai'

Ae'*

Ai'*

triangles

SEARCH value

center on K005

comments

1

8

-70.58390235756

X5/E498

on X1-X18

-43.97562105826

X(3467) = E491

on X1-X195

-35.17663270180

X(3471) = (#O)*

on X4-X195, isogonal of the tangential of O

0

9

-30.55984867257

2

5

-25.54564173887

X5/X61

on X4-X18 and X54-X61

4

8

-25.54564173887

X5/X61

on X4-X18 and X54-X61

7

11

-25.54564173887

X5/X61

on X4-X18 and X54-X61

2

3

-19.15788099913

X195

X5/X3

7

8

-19.15788099913

X195

X5/X3

4

11

-17.90134341250

0

6

-16.28067399778

X18

-11.96959001257

E501=E500*

on X3-X18

1

11

-10.89453772135

-9.21852338354

 

on X1-X628

-8.75340796080

(X5/X18)*

-6.18775905247

X54

2

4

-6.03883592346

X(3462) = E502 = X5/X4

on X4-X54 and X195-X2121

0

3

-5.67661941093

X4

6

9

-5.65363167559

5

11

-5.64425182639

3

8

-5.29953617839

 

on X3-X18

7

10

-5.29953617839

 

on X3-X18

8

11

-4.58243757196

-3.90988152470

(X5/E560)*

-3.70747719830

X5/E499

-2.90770500510

(X5/E500)*

-2.34250300590

(X5/E499)*

1

10

-2.33728658165

-2.21235552520

X5/E560

-2.08848250120

(X5/E491)*

6

12

-1.95442776483

-1.64508530484

E499=E498*

-1.34076932689

X(3463) = E503 = E502*

on X3-X2120

-0.78316288550

 

on X1-X627

-0.62737826095

X(3489) = E560=X627*

on X5-X627

-0.60900246355

(#H)*

isogonal of the tangential of H

8

10

-0.50252073688

5

10

-0.38889052256

0

8

-0.37470151131

X62

3

5

-0.37470151131

X62

6

10

-0.37470151131

X62

-0.30754962330

X(3459) = E557 = X195*

on X4-X2120 and X5-X195

-0.24819038319

X5/E503

1

3

-0.13439182998

X(3336) = E490

on X1-X3

-0.09644156557

(X5/E498)*

-0.01673252978

X5/E561

3

7

-0.00396742485

 

on X3-X17

8

9

-0.00396742485

 

on X3-X17

4

5

0.00380995175

6

11

0.00380995175

0.01423970330

X(3490) = E561=X628*

on X5-X628

2

6

0.03185131329

X5/X62

on X4-X17 and X54-X62

4

7

0.03185131329

X5/X62

on X4-X17 and X54-X62

8

12

0.03185131329

X5/X62

on X4-X17 and X54-X62

0.04216393294

X5/E557

1

4

0.05065713778

X(3468) = X5/X3336

on X1-X4

7

12

0.34430234915

1

7

0.49615812266

X5/E500

on X1-X17

2

9

0.49936417563

11

12

0.50243416641

#H

tangential of H

0

2

0.55287174163

X5

3

4

0.55287174163

X5

5

7

0.55287174163

X5

6

8

0.55287174163

X5

9

11

0.55287174163

X5

10

12

0.55287174163

X5

1

5

0.68838123821

E500

on X1-X61

2

12

0.69363920522

10

11

0.78847648057

0.86588575053

X2121

X2120*

0.90047708562

(X5/X61)*

0

11

0.94522557017

9

10

0.97735900985

X(3470) = #O

tangential of O

1.12427146000

(X5/E501)*

1

12

1.29989476029

2

10

1.31531857256

1.32753457700

X(3461) = (X5/X1)*

0

5

1.38665455007

X17

0

1

1.69030850946

X1

4

6

1.81224721118

5

12

1.81224721118

7

9

1.84130663013

1.87961569030

X2120

X5/X54

2

11

1.89737799592

0

7

1.92170828086

X61

3

6

1.92170828086

X61

5

9

1.92170828086

X61

1

9

1.99862864623

2.00016007300

X5/E501

2.09841058440

(X5/X17)*

0

10

2.35123193553

9

12

3.10001130310

3

12

5.11479727733

X627

4

9

5.11479727733

X627

6

7

5.11479727733

X627

1

6

5.27357128758

E498

on X1-X62

5.44356595290

X5/E491

5.58159639350

(X5/E503)*

0

4

6.78236289420

X3

5

6

6.78236289420

X3

3

9

7.08392671609

4

12

7.34387740579

2

8

7.63837849959

X5/X18

3

10

8.53849280298

9.36263622100

(X5/X62)*

10.37403549600

(X5/E557)*

0

12

11.80585277728

2

7

14.69984666597

X5/X17

16.26406858488

X(3469) = (X5/E490)*

3

11

40.44945437302

X628

4

10

40.44945437302

X628

5

8

40.44945437302

X628

98.67000462400

(X5/E561)*

1

2

113.96312522690

X(3460) = X5/X1

on X1-X54