Darboux 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.

Given a triangle and a point , the pedal triangle of is formed by the feet of the perpendiculars from to the three sides of triangle .


The Darboux cubic of (orange curve) is the set of all positions of such that its pedal triangle is in perspective with . In other words, the three gray dashed lines meet at a single point if and only if the point is on the cubic.

Let , , be the side lengths and let , , be the excenters of .

Then the equation of the Darboux cubic of in barycentric coordinates is given by

, where the sum is over all six permutations of the variables , , .

The Darboux cubic passes through the points , , and the Kimberling centers , , , , , , [1].

You can drag the vertices , , and the point .


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




[1] C. Kimberling, "Encyclopedia of Triangle Centers." http://faculty.evansville.edu/ck6/encyclopedia.

[2] B. Gilbert. "K004 Darboux Cubic = pk (X6, X20)." (Aug 2, 2022) bernard-gibert.pagesperso-orange.fr/Exemples/k004.html.

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.