Thomson Cubic

Initializing live version
Download to Desktop

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

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


Given a triangle , the Thomson cubic of 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 , , be the side lengths of the reference triangle and , , be the excenters. Then the equation of the Thomson cubic of triangle in barycentric coordinates is

, where indicates that the sum is taken over all six permutations of , , .

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

, , , , , , , , , , , , , , , , , [3].

You can drag the vertices , and .


Contributed by: Minh Trinh Xuan (August 2022)
Open content licensed under CC BY-NC-SA




[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)

[3] Encyclopedia of Triangle Centers (ETC).

[4] B. Gilbert. "Catalogue of Triangle Cubics." (Aug. 3, 2022)

Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.