Home page | Catalogue | Classes | Tables | Glossary | Notations | Links | Bibliography | Thanks | Downloads | Related Curves

See the related K397 for another equation

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

four foci of the MacBeath inconic : X(3), X(4) and two imaginary

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 (at Q) if and only if P lies on the Napoleon cubic (together with the line at infinity). H, P, Q are collinear and the locus of Q is Q093. Compare with K001, property 5.
  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 lies 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.
  7. Let P be a variable point. Reflect B in PC, C in PB and denote by La the perpendicular bisector of these two points. Lb, Lc are defined cyclically. La, Lb, Lc concur (at Q) if and only if P lies on K005, together with the circumcircle and the line at infinity. (Ngo Quang Duong, ADGEOM #3309). The locus of Q is Q114, together with the Euler line and the Jerabek hyperbola.

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 collinear (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 collinear 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/X6192

on X1-X18

-43.97562105826

X3467

on X1-X195

-35.17663270180

X3471

on X4-X195, isogonal of the tangential of O

0

9

-30.55984867257

on X3-X628

2

5

-25.54564173887

X8839 = X5/X61

on X4-X18 and X54-X61

4

8

7

11

2

3

-19.15788099913

X195

X5/X3

7

8

4

11

-17.90134341250

0

6

-16.28067399778

X18

-11.96959001257

X7345

 

1

11

-10.89453772135

-9.21852338354

 

on X1-X628

-8.75340796080

(X5/X18)*

-6.18775905247

X54

2

4

-6.03883592346

X3462 = 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

X8930

on X3-X18

7

10

8

11

-4.58243757196

on X61-X3490

-3.90988152470

(X5/X3489)*

-3.70747719830

X5/X7344

-2.90770500510

(X5/X6191)*

-2.34250300590

(X5/X7344)*

1

10

-2.33728658165

-2.21235552520

X5/X3489

-2.08848250120

(X5/X3467)*

6

12

-1.95442776483

-1.64508530484

X7344 =X6192*

-1.34076932689

X3463 = X3462*

on X3-X2120

-0.78316288550

 

on X1-X627

-0.62737826095

X3489 = 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

6

10

-0.30754962330

X3459 = X195*

on X4-X2120 and X5-X195

-0.24819038319

X5/X3463

1

3

-0.13439182998

X3336

on X1-X3

-0.09644156557

(X5/X6192)*

-0.01673252978

X5/X3490

3

7

-0.00396742485

X8929

on X3-X17

8

9

4

5

0.00380995175

X8919

on X4-X61

6

11

0.01423970330

X3490 = X628*

on X5-X628

2

6

0.03185131329

X8837 = X5/X62

on X4-X17 and X54-X62

4

7

8

12

0.04216393294

X5/X3459

1

4

0.05065713778

X3468 = X5/X3336

on X1-X4

7

12

0.34430234915

1

7

0.49615812266

X5/X6191

on X1-X17

2

9

0.49936417563

#X17

tangential of X17

11

12

0.50243416641

#H = X5/X3470

tangential of H

0

2

0.55287174163

X5

3

4

5

7

6

8

9

11

10

12

1

5

0.68838123821

X6191

on X1-X61

2

12

0.69363920522

10

11

0.78847648057

0.86588575053

X2121

X2120*

0.90047708562

(X5/X61)*

0

11

0.94522557017

#X61

tangential of X61

9

10

0.97735900985

X3470

tangential of O

1.12427146000

(X5/X7345)*

1

12

1.29989476029

2

10

1.31531857256

#X18

tangential of X18

1.32753457700

X3461 = (X5/X1)*

0

5

1.38665455007

X17

0

1

1.69030850946

X1

4

6

1.81224721118

X8918

on X4-X62

5

12

7

9

1.84130663013

1.87961569030

X2120

X5/X54

2

11

1.89737799592

0

7

1.92170828086

X61

3

6

5

9

1

9

1.99862864623

2.00016007300

X5/X7345

2.09841058440

(X5/X17)*

0

10

2.35123193553

9

12

3.10001130310

3

12

5.11479727733

X627

4

9

6

7

1

6

5.27357128758

X6192

on X1-X62

5.44356595290

X5/X3467

5.58159639350

(X5/X3463)*

0

4

6.78236289420

X3

5

6

3

9

7.08392671609

4

12

7.34387740579

2

8

7.63837849959

X5/X18

on X18-X54

3

10

8.53849280298

9.36263622100

(X5/X62)*

10.37403549600

(X5/X3459)*

0

12

11.80585277728

#X62

tangential of X62

2

7

14.69984666597

X5/X17

on X17-X54

16.26406858488

X3469 = X3468*

3

11

40.44945437302

X628

4

10

5

8

98.67000462400

(X5/X3490)*

1

2

113.96312522690

X3460 = X5/X1

on X1-X54