TOPICS
Search

Homogeneous Barycentric Coordinates

Homogeneous barycentric coordinates are barycentric coordinates normalized such that they become the actual areas of the subtriangles. Barycentric coordinates normalized so that

t_1+t_2+t_3=1,
(1)

so that the coordinates give the areas of the subtriangles normalized by the area of the original triangle are called areal coordinates (Coxeter 1969, p. 218). Barycentric and areal coordinates can provide particularly elegant proofs of geometric theorems such as Routh's theorem, Ceva's theorem, and Menelaus' theorem (Coxeter 1969, pp. 219-221).

The homogeneous barycentric coordinates corresponding to exact trilinear coordinates (a^':b^':c^') are (t_1,t_2,t_3), where

t_1=1/2aa^'
(2)
t_2=1/2bb^'
(3)
t_3=1/2cc^'.
(4)

The homogeneous barycentric coordinates for some common triangle centers are summarized in the following table, where R is the circumradius of the reference triangle.

triangle centerhomogeneous barycentric coordinates
circumcenter O(1/2aRcosA,1/2bRcosB,1/2cRcosC)
incenter I((a^2bc)/(4(a+b+c)R),(ab^2c)/(4(a+b+c)R),(abc^2)/(4(a+b+c)R))
orthocenter H(aRcosBcosC,bRcosAcosC,cRcosAcosB)
symmedian point K((a^3bc)/(4(a^2+b^2+c^2)R),(ab^3c)/(4(a^2+b^2+c^2)R),(abc^3)/(4(a^2+b^2+c^2)R))
triangle centroid G((abc)/(12R),(abc)/(12R),(abc)/(12R))

See also

Areal Coordinates, Barycentric Coordinates, Trilinear Coordinates

Explore with Wolfram|Alpha

References

Coxeter, H. S. M. "Barycentric Coordinates." §13.7 in Introduction to Geometry, 2nd ed. New York: Wiley, pp. 216-221, 1969.

Referenced on Wolfram|Alpha

Homogeneous Barycentric Coordinates

Cite this as:

Weisstein, Eric W. "Homogeneous Barycentric Coordinates." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HomogeneousBarycentricCoordinates.html

Subject classifications