Neuberg Cubic

Given a triangle

ABC

, the Neuberg cubic is the set of all points

P

whose reflections in the sides

BC

,

CA

and

AB

form a triangle perspective to

ABC

. It is a self-isogonal cubic with pivot point at the Euler infinity point

X

30

[1]. The name comes from the geometer Joseph Jean Baptiste Neuberg for his 1894 paper.

Let

a

,

b

,

c

be the side lengths of

ABC

and let

I

a

,

I

b

,

I

c

be the excenters of

ABC

. Then the equation of the Neuberg cubic of

ABC

in barycentric coordinates

x:y:z

is given by

∑

cyc



2

a

(

2

b

+

2

c

)+

2

(

2

b

-

2

c

)

-2

4

a

x

2

c

2

y

-

2

b

2

z

=0

, where the cyclic sum is over all six permutations of

x

,

y

,

z

.

The cubic passes through the points

I

a

,

I

b

,

I

c

and the Kimberling centers

X

1

,

X

3

,

X

4

,

X

13

,

X

14

,

X

16

,

X

30

,

X

74

,

X

370

,

X

399

,

X

484

,

X

616

,

X

616

.

You can drag the vertices

A

,

B

and

C

.

References

[1] C. Kimberling. "Encyclopedia of Triangle Centers." (Aug 2, 2022) faculty.evansville.edu/ck6/encyclopedia.

[2] B. Gilbert. "Catalogue of Triangle Cubics." (Aug 3, 2022) https://bernard-gibert.pagesperso-orange.fr/ctc.html.

External Links

Relating Trilinear and Tripolar Coordinates for a Triangle

Neuberg Cubic (Wolfram MathWorld)

Euler Infinity Point (Wolfram MathWorld)

Self-Isogonal Cubic (Wolfram MathWorld)

Pivot Point (Wolfram MathWorld)

Permanent Citation

Minh Trinh Xuan

"Neuberg Cubic"
http://demonstrations.wolfram.com/NeubergCubic/
Wolfram Demonstrations Project
Published: August 4, 2022