The Van Obel theorem may serve as a linP between Ceva's Theorem and barycentric coordinates. (Another proof of Van Obel's theorem builds on the ratios of areas formed by concurrent cevians in a triangle.)

Theorem

For three cevians meeting inside ABC at point P, the following is true:

From the mechanical point of view, van Obel's theorem asserts a recombination property of the center of mass. The center of mass of three points can be computed in two ways: first, as the average of the three points and the location of the cumulative mass. Second, as the center of mass of one of the three points (A in the diagram) and the center of mass of the other two (B and C).

The proof is very simple and follows easily from juxtaposition of three pairs of similar triangles: PHG and PBC, AEG and EBC, and AFH and BCF.

For the following I just have to simplify notations. Which actually means introducing a lot of new symbols. So, let u = PD/AD, v = EP/EB, w = FP/FC, and CD/DB = r_a, AE/EC = r_b, BF/FA = r_c. Ceva's theorem says that

Van Obel's theorem is equivalent to 1/u = r_b + 1/r_c + 1. Which, from (2), is the same as

Similarly,

and

Therefore

With one additional application of the life saving (2), the latter simplifies to u + v + w = 1, as expected. Are these the barycentric coordinates of point P? Yes, they are. In (3)-(5), u, v, and w are connected exactly in the manner described at the end of the discussion on Ceva's Theorem; so that u = w_A, v = w_B, and w = w_C. Also note that from the definition u = PD/AD. It follows that u = area(PBC)/area(ABC) and similarly for v and w. The barycentric coordinates of a point P inside ABC are proportional to the areas of triangles P forms with the three sides of the triangle.

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. Simultaneous Generalization of the Theorems of Ceva and Menelaus
   • Theorems of Ceva and Menelaus, an Illustrated Generalization
10. Three glasses puzzle
11. Van Obel Theorem and Barycentric Coordinates

Copyright © 1996-2009 Alexander Bogomolny