Van Obel Theorem and Barycentric coordinates
The Van Obel theorem may serve as a link 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 ΔABC at point P, the following is true:
(1) | AP/PD = AE/EC + AF/FB |
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
(2) | r_{a}r_{b}r_{c} = 1. |
Van Obel's theorem is equivalent to 1/u = r_{b} + 1/r_{c} + 1. Which, from (2), is the same as
(3) | 1/u = r_{b} + r_{a}r_{b} + 1 |
Similarly,
(4) | 1/v = r_{c} + 1/r_{a} + 1 = r_{c} + r_{c}r_{b} + r_{a}r_{b}r_{c} = r_{c}(1 + r_{b} + r_{a}r_{b}) |
and
(5) |
1/w = r_{a} + 1/r_{b} + 1 = r_{a} + r_{a}r_{c} + r_{a}r_{b}r_{c} = r_{a}(1 + r_{c} + r_{b}r_{c}) = r_{a}(r_{a}r_{b}r_{c} + r_{c} + r_{b}r_{c}) = r_{a}r_{c}(1 + r_{b} + r_{a}r_{b}) |
Therefore
(6) | u + v + w = (1 + 1/r_{c} + r_{b})/(1 + r_{b} + r_{a}r_{b}) |
With one additional application of the life saving (2), the latter simplifies to
Barycenter and Barycentric Coordinates
- 3D Quadrilateral - a Coffin Problem
- Barycentric Coordinates
- Barycentric Coordinates: a Tool
- Barycentric Coordinates and Geometric Probability
- Ceva's Theorem
- Determinants, Area, and Barycentric Coordinates
- Maxwell Theorem via the Center of Gravity
- Bimedians in a Quadrilateral
- Simultaneous Generalization of the Theorems of Ceva and Menelaus
- Three glasses puzzle
- Van Obel Theorem and Barycentric Coordinates
- 1961 IMO, Problem 4. An exercise in barycentric coordinates
- Centroids in Polygon
- Center of Gravity and Motion of Material Points
- Isotomic Reciprocity
- An Affine Property of Barycenter
- Problem in Direct Similarity
- Circles in Barycentric Coordinates
- Barycenter of Cevian Triangle
- Concurrent Chords in a Circle, Equally Inclined
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Copyright © 1996-2018 Alexander Bogomolny
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