WOLFRAM NOTEBOOK

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

Permanent Citation

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