Darboux Cubic

Given a triangle

ABC

and a point

P

, the pedal triangle of

P

is formed by the feet of the perpendiculars from

P

to the three sides of triangle

ABC

.

The Darboux cubic of

ABC

(orange curve) is the set of all positions of

P

such that its pedal triangle is in perspective with

ABC

. In other words, the three gray dashed lines meet at a single point if and only if the point

P

is on the cubic.

Let

a

,

b

,

c

be the side lengths

ABC

and let

I

a

,

I

b

,

I

c

be the excenters of

ABC

.

Then the equation of the Darboux cubic of

ABC

in barycentric coordinates

x:y:z

is given by

∑

cyc

2

2

a

(

2

b

+

2

c

)+

2

(

2

b

-

2

c

)

-3

4

a

x

2

c

2

y

-

2

b

2

z

=0

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

x

,

y

,

z

.

The Darboux cubic passes through the points

I

a

,

I

b

,

I

c

and the Kimberling centers

X

1

,

X

3

,

X

4

,

X

20

,

X

40

,

X

64

,

X

84

[1].

You can drag the vertices

A

,

B

,

C

and the point

P

.

References

[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.

External Links

Pedal Triangle (Wolfram MathWorld)

Exact Trilinear Coordinates and the Pedal Triangle

Relating Trilinear and Tripolar Coordinates for a Triangle

Darboux Cubic (Wolfram MathWorld)

Permanent Citation

Minh Trinh Xuan

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