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The Grebe triangle is defined and studied here.

This page is only a compilation of various cubics and higher degree curves passing through its vertices G1, G2, G3 and other remarkable points.

 

Cubics

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

Knnn* and Knnn*G denote the isogonal transforms of Knnn with respect to ABC and the Grebe triangle respectively.

cubic

 

type

Xi on the cubic for i =

points on (O)

remarks

K102

c

isogonal pK

1, 2, 6, 43, 87, 194, 3224, 15963, 15964, 15965, 15966, 15967, 15968, 39641, 39642, 40139

 

 

K138

 

equilateral

2, 6, 5652, 14898, 14899, 35607, 35608, 35609

Thomson triangle

 

K177

c

pK

2, 3, 6, 25, 32, 66, 206, 1676, 1677, 3162, 19615, 41378, 41379, 52041

 

K141*

K281

c

spK, nodal

2, 6, 182, 996, 1001, 1344, 1345, 4846, 5967, 10002, 14356, 14609, 18775, 45998, 46023, 46024, 51510

 

K280*

K642

c

isog. pK wrt G1G2G3

4, 206, 1676, 1677

 

 

K643

c

spK, stelloid

4, 6, 4846, 8743, 39641, 39642

 

see note 2

K644

c

pK

2, 4, 6, 83, 251, 1176, 1342, 1343, 8743, 40357, 40358, 40404

 

K836*

K729

c

spK

2, 6, 1383, 39162, 39163, 39164, 39165, 51797

 

K287*

K731

c

spK

6, 83, 39641, 39642

 

 

K835

c

spK

3, 4, 6, 32, 1995, 3425, 8743, 14356

 

K527*

K1161

 

stelloid

3, 6, 40122

CircumTangential triangle

 

K1241

 

central

3, 5, 182, 206, 5092, 44882, 44883, 44884, 44885

antipodes G1, G2, G3

K644*G

K1249

 

 

2, 3, 22, 7712, 46264

antipodes of (O) ∩ K006

 

K1286

 

focal

2, 3, 23, 110, 182, 187, 353, 9829, 11645, 15080, 39162, 39163, 39164, 39165, 51797, 51798, 51799, 51800

X(110), J1, J2

 

K1292

 

stelloid

1, 6, 20, 194, 35237, 46264

antipodes of (O) ∩ K006

K1291*G

K1317

c

 

2, 4, 32, 194, 1340, 1341, 1383, 9463, 14609, 45096, 52672, 52765

 

K1316*

 

c

spK+

1, 6, 996

 

see note 2

 

c

spK+

6, 194, 3224

 

see note 2

Note 1 : a pK passes through G1, G2, G3 if and only if its pole, pivot, isopivot lie on K1258 = pK(X251 x X32, X251), K644, K177 respectively.

Note 2 : any spK(P, Q = midpoint of X6,P) passes through G1, G2, G3 and X6. It also contains the infinite points of pK(X6, P) and the foci of the inconic with center Q. Its isogonal transform is spK(X6, Q). See CL055.

This spK passes through P when P lies on the Grebe cubic K102.

This spK is a K+ if and only if P lies on the circular cubic K837 passing through X(1), X(3), X(147), X(194), X(511), X(2930), X(7772). K643 is the most remarkable example obtained with P = X(3). Two other spK+ are listed with P = X(1), X(194), two points on K102.

 

Higher degree curves

Q019 = Q094*, Q138, Q142, Q144, Q158 = Q157*.