Barycentric Coordinates: A Tool

Assume that relative to ABC, a given point P has barycentric coordinates (a, b, g). The cevians AP, BP, CP crosses BC, AC and AB in A', B' and C', respectively. From Ceva's theorem the product of the ratios AC'/C'B · BA'/A'C · CB'/B'A = 1. These ratios are bound to be related to the barycentric coordinates of P, but how? And the other way round: given the three ratios, do they determine the coordinates a, b, g of P? The answer is, Yes, they do; we'll see below how.

By definition, a, b, g can be treated as weights placed at the vertices A, B, C so that P becomes the center of gravity of the three material points:

The formula simpifies if we assume

Rewrite (1) as

Which tells us that the point K = b/(b + g)·B + g/(b + g)·C lies on BC whereas P belongs to AK. In our notations, K = A' and

Or

It follows that

A', therefore, is the center of gravity of the material points B and C with weights b and g.

The identity (3) provides a step-by-step construction of the barycenter of three material points. First, find the barycenter of any two points and place there their combined weight. Next, find the barycenter of the so obtained and the remaining third point.

(4) answers one question: the cevians through point P with the barycentric coordinates (a, b, g) cut the sides BC, AC, and AB in the ratios

Conversely, assume that the points A', B', C' are given so that

with lmn = 1. What are the barycentric coordinates a, b, g of point P whose existence is implied by Ceva's theorem?

We are looking for a, b, g, which from (5) and (6) satisfy

Recollect that the barycentric coordinates are homogeneous and in the absence of additional restrictions are not unique. With this in mind, set in the first equation in (7) b = 1. This forces g = l. From the second equation then we get a = gm = lm. Finally, from the third equation b = lmn, which we observe does not contradict b = 1 because of the Ceva's condition lmn = 1.

P therefore has the coordinates

Note that these are exactly the expressions that appear in Routh's theorem, except that now the three are equal due to the Ceva's condition lmn = 1.

The applet below is a simple tool that demonstrates the connection of the barycentric coordinates of pPoint P relative to ABC and the ratios in which the cevians through P split the side of the triangle. The applet represents three of the many ways in which these ratios can be written.

Barycenter and Barycentric Coordinates

1. 3D Quadrilateral - a Coffin Problem
2. Barycentric Coordinates
3. Barycentric Coordinates: a Tool
4. Barycentric Coordinates and Geometric Probability
   • Stick Broken Into Three Pieces (Trilinear Coordinates)
   • Stick Broken Into Three Pieces (Cartesian Coordinates)
5. Ceva's Theorem
6. Determinants, Area, and Barycentric Coordinates
7. Maxwell Theorem via the Center of Gravity
8. Medians in a Quadrilateral
9. Three glasses puzzle
10. Van Obel Theorem and Barycentric Coordinates

Copyright © 1996-2009 Alexander Bogomolny