Basic Parameters of the Kimberling Center X(49)
Inscribe two triangles and in a reference triangle such that[more]
Then the triangles and are both inscribed in a circle known as the sine-triple-angle circle (shown in red), whose center is .
, , 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 , , .[less]
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.
 C. Kimberling. "Encyclopedia of Triangle Centers." (May 23, 2023) faculty.evansville.edu/ck6/encyclopedia.