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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 .[more]
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 , , , , , , .
You can drag the vertices , , and the point .[less]
Contributed by: Minh Trinh Xuan (August 2022)
Open content licensed under CC BY-NC-SA
 C. Kimberling, "Encyclopedia of Triangle Centers." http://faculty.evansville.edu/ck6/encyclopedia.
 B. Gilbert. "K004 Darboux Cubic = pk (X6, X20)." (Aug 2, 2022) bernard-gibert.pagesperso-orange.fr/Exemples/k004.html.