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The Thomson triangle T is defined and studied here.

This page is only a compilation of various cubics and higher degree curves passing through its vertices Q1, Q2, Q3 and other remarkable points.

See also Table 54, column P = [X2].

 

Cubics

c denotes a circum-cubic, otherwise the three remaining points on the circumcircle (O) are mentioned.

Knnn* and Knnn*T denote the isogonal transforms of Knnn with respect to ABC and T respectively.

cubic

 

Type

Xi on the curve for i =

points on (O)

remarks

K002

c

pK

see the page

 

 

K078

 

stelloid

1, 2, 3, 165, 5373, 6194

CircumTangential triangle

K003 in T

K138

 

equilateral

2, 6, 5652

Grebe triangle

 

K167

c

pK

3, 6, 3167, 8770

 

K181*

K172

c

pK

3, 6, 25, 55, 56, 64, 154, 198, 1033, 1035, 1436, 7037

 

K007*

K280

c

spK, nodal

2, 6, 262, 378, 995, 1002, 1340, 1341, 5968, 7757

 

K281*

K297

c

nodal

3, 6, 183, 956, 1344, 1345, 3445, 5968, 9717

 

K295*

K463

 

focal

2, 3, 15, 16, 30, 110, 5463, 5464

X110, circular points at ∞

K187 in T

K581

c

spK, stelloid

2, 3, 4, 262

 

 

K609

 

 

1, 2, 3, 20

see note 1

 

K615

c

spK

2, 3, 4, 64, 154, 3424, 5373

 

K047*

K624

c

nK0+

6, 523, 2574, 2575, 5968, 8105, 8106

 

 

K625

c

nK0

6, 187, 511, 523, 690, 6137, 6138

 

 

K626

c

nodal

3, 25, 1073, 1384, 1617, 3167, 3420, 3426, 9717

 

K616*

K703

 

nK wrt T

 

antipodes of A, B, C

 

K727

 

 

2, 3, 7712

CircumTangential triangle

 

K758

 

central

2, 3, 154, 165, 376, 3576

antipodes of Q1, Q2, Q3

K002*T

K759

c

spK

2, 3, 4, 3431, 7607, 9716, 9717

 

K762*

K760

c

pK

1, 6, 9, 56, 84, 165, 198, 365

 

K308*

K761

c

pK

1, 6, 9, 55, 259, 3158, 3445

 

K365*

K764

 

central

2, 3, 6, 376, 1350, 5373, 9740, 9741

antipodes of Q1, Q2, Q3

K004 in T

K765

 

 

2, 3, 3524, 5024, 5373, 5646

points on an isogonal nK0

K002 in T

K804

c

spK

2, 3, 4, 3167, 7612

 

 

K810

c

spK

2, 3, 4, 3426

 

 

K833

 

central stelloid

2, 3, 381

reflection of T about G

 

K834

 

circular

2, 3, 30, 110, 5373, 10620

X110, circular points at ∞

K001 in T

K903

 

strophoid

3, 23, 110, 187, 6141, 6142

X110, circular points at ∞

 

K912

 

focal

3, 15, 16, 23, 110, 5663, 13858, 13859

X110, circular points at ∞

 

K913

 

circular

1, 2, 3, 30, 110, 147, 399, 5536

X110, circular points at ∞

 

Note 1 : K609 meets (O) again at three points (other than X74) on a rectangular hyperbola which is the image of the Jerabek hyperbola under the translation that maps H onto O.

Note 2 : Any pK passing through Q1, Q2, Q3 must have its pole on K346, its pivot on K002, its isopivot on K172.

Note 3 : spK(P, Q = midpoint of G,P) passes through Q1, Q2, Q3, G, P*, the infinite points of pK(X6, P), the foci of the inconic with center Q. See CL055.

 

Higher degree curves

Qnnn* is the isogonal transform of Qnnn with respect to ABC.

curve

type

Xi on the curve for i =

remarks

Q002

circular quartic

1, 3, 6, 15, 16, 358, 1135, 1137, 1155, 2574, 2575, 10221

Q003*

Q011

symgonal circular quintic

3, 1145, 1312, 1313, 1511, 2028, 2029, 2446, 2447, 5976

 

Q037

inversible bicircular quintic

1, 3, 15, 16, 30, 36, 5000, 5001

Q030*

Q062

quartic

2, 6, 523

 

Q067

bicircular quintic

1, 3, 15, 16, 30

 

Q068

circular equilateral quintic

1, 3, 164, 399, 2448, 2449

 

Q069

circular quintic

1, 3

 

Q071

bicircular quintic

3, 30, 187, 468, 1155, 1319

Q001*

Q076

sextic

1, 2, 6, 13, 14, 15, 16, 110, 523

 

Q090

nodal quartic

2, 6, 15, 16, 55, 385, 672, 5638, 5639

Q012*

Q091

quintic

1, 3, 5373

 

Q095

quintic

2, 13, 14, 99

 

Q097

bicircular quintic

3, 15, 16, 36, 55, 187

Q096*

Q098

inversible circular quartic

3, 6, 187, 2574, 2575, 3513, 3514, 5000, 5001

Q044*

Q101

unicursal quartic

6, 325, 1515, 2574, 2575, 8779

 

Q113

circular quartic

1, 3, 6, 64, 2574, 2575, 5373

Q063*

Q118

circular quartic

2, 376, 3413, 3414, 5000, 5001

 

Q136

circular quartic

3, 4, 6, 15, 16, 23, 2574, 2575, 7712

Q050*

Q137

circular quartic

3, 6, 15, 16, 2574, 2575

Q135*