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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(W1, 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

Note : W1 = a^3(b+c-a)[bc-a(b+c-a)] : ... : ... = X(2)X(1252) /\ X(55)X(2195).

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.