Thomson Cubic

A circumconic is a conic section through the vertices of a triangle[1].

Given a triangle

ABC

, the Thomson cubic of

ABC

is the set of the centers of circumconics whose normals at the vertices are concurrent[2]. It is a self-isogonal cubic with pivot point at the triangle centroid.

Let

a

,

b

,

c

be the side lengths of the reference triangle

ABC

and

I

a

,

I

b

,

I

c

be the excenters. Then the equation of the Thomson cubic of triangle

ABC

in barycentric coordinates

(x,y,z)

is

∑

cyc

x

2

c

2

y

-

2

b

2

z

=0

, where

cyc

indicates that the sum is taken over all six permutations of

x

,

y

,

z

.

The solution is traced in red. Some of the Kimberling centers on the Thomson cubic are:

I

a

,

I

b

,

I

c

,

X

1

,

X

2

,

X

3

,

X

4

,

X

6

,

X

9

,

X

57

,

X

223

,

X

282

,

X

1073

,

X

1249

,

X

3341

,

X

3342

,

X

3343

,

X

3344

[3].

You can drag the vertices

A

,

B

and

C

.

References

[1] C. Kimberling, "Triangle Centers and Central Triangles." Congressus Numerantium, 129, 1–295, 1998.

[2] B. Gilbert. "Thomson Cubic = pK(X6,X2)." (Jul 20, 2022) bernard-gibert.pagesperso-orange.fr/Exemples/k002.html.

[3] Encyclopedia of Triangle Centers (ETC). https://faculty.evansville.edu/ck6/encyclopedia/etc.html.

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

External Links

Circumconic (Wolfram MathWorld)

Self-Isogonal Cubic (Wolfram MathWorld)

Relating Trilinear and Tripolar Coordinates for a Triangle

Thomson Cubic (Wolfram MathWorld)

Permanent Citation

Minh Trinh Xuan

"Thomson Cubic"
http://demonstrations.wolfram.com/ThomsonCubic/
Wolfram Demonstrations Project
Published: August 3, 2022