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

Let P be a point in the plane of the reference triangle ABC and denote by

cevP : the cevian triangle of P
acvP : the anticevian triangle of P
pedP : the pedal triangle of P
apdP : the antipedal triangle of P
ccvP : the circumcevian triangle of P
cacP : the circumanticevian triangle of P
refP : the reflection triangle of P in the sidelines of ABC
symP : the reflection triangle of P in the vertices of ABC.
cevP* : the cevian triangle of the isogonal conjugate P* of P, and similarly for the other triangles

The following tables give the loci of P such that two of these triangles are perspective. Note the frequent occurence of the Darboux cubic K004.

true means the triangles are perspective for any P, true(H) means the triangles are actually homothetic

L denotes the line at infinity, C denotes the circumcircle, 6B the union of the six bisectors.

 

ABC

cevP

acvP

pedP

apdP

ccvP

cacP

refP

cevP

true

 

 

 

 

 

 

 

acvP

true

true

 

 

 

 

 

 

pedP

K004

union of the altitudes

K004

 

 

 

 

 

apdP

K004

K004

C, 6B

K004

 

 

 

 

ccvP

true

true

true

L, C and K003

true

 

 

 

cacP

union of the symmedians

union of the symmedians

true

a nonic

L, C and the sextic Q028

true

see

Table 34

 

 

refP

K001

K060

true

true(H)

L, C and K006

C and K003

a nonic

 

symP

true(H)

true

true

L and K243

L, C and K001

true

L and the quintic Q029

L and K004

 

cevP

acvP

pedP

apdP

ccvP

cacP

refP

symP

cevP*

6B

true

three conics, see Table 11

L, C and K004

C and a septic

C and a septic

a sextic

L and a quintic

acvP*

true

6B

K004

C and 6B

C and Q026

K102 and a nK(X6,X2)

Q075

L and a quintic

pedP*

three nodal cubics, see Table 11

K004

three isogonal focal pK

true(H)

C and 10th degree

12th degree

a nonic

L and a circular septic

apdP*

L, C and K004

L, C and 6B

true(H)

L, C and 6B

L, C and K003

L, C and a sextic

true(H)

L and a circular quintic

ccvP*

L and an octic

L and a quintic

L and an octic

L, C and K003

C, 6B and K024

L and a quintic

L and an octic

L and a sextic

cacP*

a quintic

K102 and a nK(X6,X2)

a nonic

L, C and a sextic

C and Q026

6B and a sextic

a nonic

L and an octic

refP*

a nonic

Q075

a nonic

true(H)

C and 10th degree

12th degree

K006

L and a bicircular septic

symP*

C and a septic

C and a septic

C and a quintic

L, C and a septic

L and a circular sextic

C and 10th degree

C and Q067

true(H)

 

Loci related to refP perspective with a fixed triangle T

This section written with Martin Acosta's cooperation

In this section we study the locus L(P) of P such that refP is perspective with a FIXED given triangle T and, if possible, the locus L(Q) of the perspector Q. Several loci are already available in the table above when T is the reference triangle ABC.

T is the anticevian triangle

Let T = A1B1C1. Denote by A2, B2, C2 the reflections of A1 in BC, B1 in CA, C1 in AB respectively. Let A3 = BC2 /\ CB2 and define B3, C3 similarly. The locus L(P) is a circular circum-cubic which contains M, A2, B2, C2, A3, B3, C3.

L(P) also contains A1 if and only if M lies on the isogonal pK with pivot the midpoint of the altitude AH. Hence, L(P) contains the three points A1, B1, C1 if and only if M is an in/excenter. When M = I, the cubic is K269 = pK(X6, X515) and when M is an excenter, it is the isogonal pK whose pivot is the corresponding extraversion of X(515), the infinite point of the line IH.

L(P) is a K0 (i.e. a cubic without term in xyz) if and only if M lies on the Napoleon cubic K005, and in this case, it is always a pK. This is the case of K269.

***

T is the antipedal triangle

Let T = A1B1C1. Denote by A2, B2, C2 the reflections of A1 in BC, B1 in CA, C1 in AB respectively. Let A3 = BC2 /\ CB2 and define B3, C3 similarly. The locus L(P) is a circular circum-cubic which contains M* = isogonal conjugate of M, A2, B2, C2, A3, B3, C3.

L(P) also contains A1 if and only if M lies on the isogonal pK with pivot the antipode of A on the circumcircle. Hence, L(P) contains the three points A1, B1, C1 if and only if M is an in/excenter. When M = I, the cubic is K269 = pK(X6, X515) and when M is an excenter, it is the isogonal pK whose pivot is the corresponding extraversion of X(515), the infinite point of the line IH.

L(P) is a K0 (i.e. a cubic without term in xyz) if and only if M lies on K270 = pK(X6, X1503).

 

Loci related to circumperp triangles

The circumperp triangles CP1, CP2 and their tangential triangles TCP1, TCP2 are defined in Clark Kimberling's TCCT (§§ 6.21 upto 6.23). The vertices of CP1 (resp. CP2) are the second intersections of the circumcircle with the external (resp. internal) bisectors of ABC.

The following table gives the loci of P such that a triangle related to P is perspective with one of these four triangles.

 

CP1

CP2

TCP1

TCP2

cevP

line X1-X7 and circumconic with perspector X(1)

pK(X81, X86)=K317

pK(X1252, X4998) through X(2), X(55), X(100), X(1252), X(4998)

pK(X593, X261)=K320

acvP

line X1-X6 and circumconic with perspector X(55)

pK(X1333, X81)=K319

pK(X16283, X2) through X(2), X(55), X(1376)

pK(X5019, X2)=K321

pedP

a non-circum cubic through X(1), X(3), X(1490)

a non-circum cubic through X(1), X(3)

a non-circum cubic through X(3)

a non-circum cubic through X(3)

apdP

L, C, and pK(X1333, X21)=K318

L, C and the line X1-X3

L, C and a cubic through X(4), X(56), X(945)

L, C and a cubic through X(3), X(4), X(55), X(64)

ccvP

C, external bisectors and line X1-X3

C, internal bisectors and line X36-X238

C, line X1-X3 and a cubic

C, line X36-X238 and pK(X1333, X21)=K318

cacP

internal bisectors and a cubic

external bisectors and pK(X1333, X81)=K319

line X1-X6, circumconic with perspector X(55) and a cubic

pK(X1333, X81)=K319 and another cubic

refP

a circular non-circumcubic through X(1), X(3), X(101), X(515), X(993)

a focal non-circumcubic through X(1), X(3), X(36), X(109), X(515)

a circular non-circumcubic through X(3), X(30)

a circular non-circumcubic through X(3), X(30)

symP

L and a conic through X(382), X(1482)

L and a conic through X(1), X(21), X(382), X(1482)

true

true

These triangles CP2 and CP1 can be seen as the circumcevian and circumanticevian triangles of the incenter. This latter point may be replaced by any other fixed point Q as far as it does not lie on a sideline or on the circumcircle. The most interesting generalization is obtained with ccvQ, the circumcevian triangle of Q, since we always find three related pK.

The following table gives the loci of P such that ccvQ is perspective to cevP, acvP (or cacP) and also the locus of the perspector which is the same for all these triangles.

 

pole of the pK

pivot of the pK

isopivot of the pK

cevP

p^2 / [a^2(c^2q+b^2r)] : :

p / [a^2(c^2q+b^2r)] : :

Q

acvP

a^2 / (c^2q+b^2r) : :

1 / (c^2q+b^2r) : :

K = X(6)

persp.

a^2 / (c^2q+b^2r) : :

(-a^2qr+b^2rp+c^2pq) / (c^2q+b^2r) : :

a^2 / (-a^2qr+b^2rp+c^2pq) : :

 

Loci related to other triangles

The following table gives another selection of loci related to several triangles as in Clark Kimberling TCCT p.155 & sq and other "classical" triangles.

Cevian and Anticevian, Pedal and Antipedal triangles of a fixed point Q

 

Cevian triangle

Anticevian triangle

Pedal triangle

Antipedal triangle

cevP

cevian lines of Q

true

3 lines

sidelines

acvP

true

cevian lines of Q

sidelines

3 lines

pedP

3 lines perpendicular to the sidelines of ABC at the vertices of the cevian triangle of Q

K004

(see note below)

3 lines

L, C and a circum-cubic

apdP

L, C and K004

L, C and 3 lines

L, C and a circum-cubic

L, C and cevian lines of Q

ccvP

C and a circum-quartic through Q

sidelines, a line and C

C and a circum-quartic

C, sidelines and a line

cacP

trilinear polar of Q and a quintic

sidelines, a line and a conic

a circum-sextic

sidelines, C and a circum-cubic

refP

a cubic

a circum-cubic

a cubic

L, C and a circular circum-cubic

symP

L and a conic

L and a conic

L and a conic

C and a conic

Note : for any Q and for any P on K004, pedP and acvQ are perspective and the locus of the perspector is a cubic which is the transform of the Lucas cubic K007 under the mapping M -> M/Q (M-Ceva conjugate of Q). This cubic is a pK in acvQ with pivot the isogonal of Q in the orthic triangle and isopivot X(69)/Q. When Q = H, it is also a pK in ABC.

Intangents and Extangents triangles

details in TCCT §6.16 p.161 and §6.17 p.162

 

Intangents triangle

Extangents triangle

cevP

Feuerbach hyperbola

K033 = pK(X37, X8)

acvP

Circum-hyperbola with perspector X(663) passing through X(6), X(9), X(19), etc

K362 = pK(X213, X1)

pedP

Central cubic with center X(1) passing through X(4), X(40), X(944)

Central cubic with center X(40) passing through X(4), X(40)

apdP

L, C and a circum-cubic through X(3), X(40), X(84)

L, C and pK(X3 x X942, X1) passing through X(1), X(3), X(58), X(500), X(501), X(942)

ccvP

C and a circum-quartic through X(4), X(84), X(365)

C and a circum-quartic through X(4), X(55), X(65), X(365)

cacP

Circum-sextic through X(2), X(57), X(365)

Circum-sextic through X(2), X(6), X(365)

refP

L and a conic through X(1), X(4), X(19), X(221)

L and a conic through X(4), X(33), X(40), X(55), X(199)

symP

L and a rectangular hyperbola through X(40), X(2574), X(2575)

L and a rectangular hyperbola through X(1), X(65), X(71), X(2574), X(2575)

Reflected triangles

A'B'C' is the triangle formed by the reflections of A, B, C in the sidelines of ABC.

The hexyl triangle is the triangle formed by the reflections of the excenters in the circumcenter O of ABC.

A1B1C1 is the triangle formed by the reflections of O in the vertices of ABC (TCCT §6.13).

 

A'B'C'

Hexyl

A1B1C1

cevP

K060

K344

a pK through X(2), X(3), X(5), X(69), X(1173), X(1994)

acvP

K005

K343

a pK with pivot G through X(2), X(3), X(6), X(1656)

pedP

K127

a cubic through X(1), X(1498)

a cubic through X(3), X(20), X(382)

apdP

K364

L, C and K343

L, C and a cubic through X(3), X(4), X(64), X(1657)

ccvP

C and a circum-quartic through H

C and a quartic through X(1), X(3), X84), X(513)

C and a circular circum-quartic through X(3), X(1173), X(2574), X(2575)

cacP

circum-sextic through K and X(1989)

a sextic through X(1), X(6), X(57)

a sextic through X(6), X(288)

refP

K060

a cubic through X(1), X(515)

a cubic through X(3), X(30), X(382), X(2080)

symP

L and a rectangular hyperbola through H, K, X(1657), X2574), X(2575)

L and a conic through X(1), X(84)

true

See other loci related to the four Brocard triangles.

See also K585, K586 for cubics related with the Morley triangle.

 

Loci related to cevian and anticevian triangles

The following table gathers together a good number of pKs, the loci of P whose cevian / anticevian triangle is perspective at Q to a certain triangle T. Note that the locus of the perspector Q is the same pK in both cases. This is a direct adaptation of César Lozada's tables in ETC, see X(8782) and X(8856), with some addional informations.

 

pKs related with cevian triangle of P

and with anticevian triangle of P

 

Triangle T

pK

Pole

Pivot

pK

Pole

Pivot

Locus of Q

1st anti-Brocard

K738

76

3978

K739

385

6

K699

anti-McCay

--

598

8785

--

8859

8860

 

1st Brocard

K322

1916

694

K128

6

385

K020

2nd Brocard

K531

3455

67

K534

3

524

K538

3rd Brocard

K532

8789

694

K128

6

385

K020

4th Brocard

K533

8791

8791

K535

25

468

K539

5th Brocard

--

3051

141

--

16285

2

pK(16285,2896)

circummedial

--

308

308

K644

251

83

K959

circumorthic

--

8794

8795

--

8882

275

K919

2nd circumperp

K317

81

86

K319

1333

81

K318

Euler

--

8796

8797

K671

53

2

pK(53,3091)

2nd Euler

--

2

317

--

216

6515

K044

5th Euler

--

8801

8801

K517

427

4

 

extangents

K033

37

8

K362

213

1

K750

2nd extouch

K007

2

69

K033

37

8

K880

3rd extouch

K007

2

69

K964

1427

7

K963

4rd extouch

--

8813

8814

--

1214

7

pK(1214,5933)

5th extouch

--

8816

8817

--

8898

7

 

Feuerbach

--

8818

3615

K672

115

1

K877

outer-Garcia

K366

321

75

K345

37

2

K033

inner-Grebe

--

494

5491

--

6421

2

 

outer-Grebe

--

493

5490

--

6422

2

 

hexyl

K344

81

8822

K343

6

63

K004

Johnson

K674

324

264

K612

216

2

K044

Kosnita

--

2

311

--

571

1994

K388

Lucas central

--

8825

588

--

8908

6

 

Lucas tangents

 

8950

493

 

8911

6

 

McCay

--

8587

8827

--

6

8859

 

midheight

K007

2

69

K004

6

20

K002

mixtilinear

--

8828

8829

--

198

57

 

3rd mixtilinear

--

7366

269

--

604

57

 

4th mixtilinear

--

1253

9

--

31

1

pK(31,165)

5th mixtilinear

K365

57

7

--

9

2

K201

6th mixtilinear

K202

1

144

K351

6

9

pK(6,2951)

Morley

K586

3275

5456

K585

6

3602

K029

inner-Napoleon

K420a

14

8836

K129a

6

395

K005

outer-Napoleon

K420b

13

8838

K129b

6

396

K005

1st Neuberg

--

3407

8840

K128

6

385

K422

2nd Neuberg

--

1916

8842

K423

6

3329

K422

orthocentroidal

K060

1989

265

K001

6

30

K005

reflection

K060

1989

265

K005

6

5

K001

1st Sharygin

K132

1

894

K673

1914

6

K960

2nd Sharygin

K323

1

239

K673

1914

6

K961

inner-squares

K070b

4

1585

K006

6

4

K424a

outer-squares

K070a

4

1586

K006

6

4

K424b

inner-Vecten

K070a

4

1586

K424b

6

3069

K006

outer-Vecten

K070b

4

1585

K424a

6

3068

K006