McCay-Griffiths 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 sides

ABC

. The circumcircle of the pedal triangle is called the pedal circle of

P

(shown in black).

The McCay–Griffiths cubic of

ABC

is the set of all points

P

such that the pedal circle of

P

is tangent to the nine-point circle (shown in green) of

ABC

. Its pivot point is the circumcenter (Kimberling center

X

3

).

Let

a

,

b

,

c

be the side lengths;

S

A

,

S

B

,

S

C

the parameters of

ABC

in Conway triangle notation and

I

a

,

I

b

,

I

c

the excenters of

ABC

.

Then the equation of the McCay–Griffiths cubic of

ABC

in barycentric coordinates

x:y:z

is

∑

cyc

2

a

S

A

x

2

c

2

y

-

2

b

2

z

=0

,

where the sum is cyclic over all six permutations of

x

,

y

,

z

.

The McCay–Griffiths cubic passes through the points

I

a

,

I

b

,

I

c

and the Kimberling centers

X

1

,

X

3

,

X

4

,

X

1075

,

X

1745

,

X

3362

,

X

13855

[1].

You can drag the vertices

A

,

B

,

C

and the point

P

.

References

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

[2] B. Gilbert. "K003 McCay Cubic=Griffiths Cubic." (Jul 29, 2022) bernard-gibert.pagesperso-orange.fr/Exemples/k003.html.

External Links

Conway Triangle Notation (Wolfram MathWorld)

Exact Trilinear Coordinates and the Pedal Triangle

Relating Trilinear and Tripolar Coordinates for a Triangle

M'Cay Cubic (Wolfram MathWorld)

Pivot Point (Wolfram MathWorld)

Barycentric Coordinates (Wolfram MathWorld)

Permanent Citation

Minh Trinh Xuan

"McCay-Griffiths Cubic"
http://demonstrations.wolfram.com/McCayGriffithsCubic/
Wolfram Demonstrations Project
Published: August 3, 2022