Basic Parameters of the Kimberling Center X(49)

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.

Inscribe two triangles and in a reference triangle such that





Then the triangles and are both inscribed in a circle known as the sine-triple-angle circle (shown in red), whose center is [1].


, , be the side lengths and ,

, , be the circumradius, inradius and semiperimeter of ,

, , be the exact trilinear coordinates of with respect to and .


&LeftBracketingBar;SubscriptBox["AX", "49"]&RightBracketingBar; =SA2-2 R2 SA2 Sω-7 R2+R2 (R4-8 S2+4 R2 Sω-Sω2)+2 S2 Sω(2 Sω-7 R2)2,

da =a3 SA (SA2-3 S2)8 S3 (7 R2-2 Sω),

dX49 =r3+6 r2 R+9 r R2+R3-3 r s27 R2-2 Sω.

You can drag the vertices , , .


Contributed by: Minh Trinh Xuan (August 25)
Open content licensed under CC BY-NC-SA


A triangle center is said to be even when its barycentric coordinates can be expressed as a function of three variables , , that all occur with even exponents. If the center of a triangle has constant barycentric coordinates, it is called a neutral center (the centroid is the only neutral center). A triangle center is said to be odd if it is neither even nor neutral.

Standard barycentric coordinates of a point with respect to a reference triangle have a sum of 1.


[1] C. Kimberling. "Encyclopedia of Triangle Centers." (May 23, 2023)


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.