Theorems of Ceva and Menelaus, an Illustrated Generalization

A theorem by M. Klamkin and A. Liu not only combines two famous theorem - those of Ceva and Menelaus - in a single statement but also extends both in a significant way.

Theorem

Let A₁A₂A₃ be a triangle with points B₁, B₂, B₃ and C₁, C₂, C₃ in sidelines A₂A₃, A₁A₃, and A₁A₂, respectively. Assume b₁, b₂, b₃ and c₁, c₂, c₃ are real numbers such that A₂B₁ = A₂A₃ / (1 + b₁), C₁A₃ = A₂A₃ / (1 + c₁) and similarly for other indices. Then C₁B₂, C₂B₃ and C₃B₁ are concurrent iff

A proof of the theorem appears in a separate page. Below is a Java applet that illustrates the theorem. You may find it useful in verifying your understanding of the barycentric coordinates.

On load, the applet displays one of the configurations known yet to the Dutch artist M. C. Escher. When the sides of ΔA₁A₂A₃ are divided into 3, 4, and 5 segments, there are twelve such concurrences whose existence is assured by the theorem.

References

1. M. S. Klamkin, A. Liu, Simultaneous Generalization of the Theorems of Ceva and Menelaus, Mathematics Magazine, Vol 65, No 1 (February 1992), pp. 48-52

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

Menelaus and Ceva

• The Menelaus Theorem
• Menelaus Theorem: proofs ugly and elegant - A. Einstein's view
• Ceva's Theorem
• Ceva in Circumscribed Quadrilateral
• Ceva's Theorem: A Matter of Appreciation
• Ceva and Menelaus Meet on the Roads
• Menelaus From Ceva
• Menelaus and Ceva Theorems
• Ceva and Menelaus Theorems for Angle Bisectors
• Ceva's Theorem: Proof Without Words
• Cevian Nest
• Cevian Triangle
• An Application of Ceva's Theorem
• Trigonometric Form of Ceva's Theorem
• Two Proofs of Menelaus Theorem
• Simultaneous Generalization of the Theorems of Ceva and Menelaus
  • Theorems of Ceva and Menelaus, an Illustrated Generalization
  

  

  Copyright © 1996-2010 Alexander Bogomolny