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

(*)	b1b2b3 + c1c2c3 + b1c1 + b2c2 + b3c3 = 1.

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.

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.

What if applet does not run?

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. Bimedians 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
12. 1961 IMO, Problem 4. An exercise in barycentric coordinates
13. Centroids in Polygon
14. Center of Gravity and Motion of Material Points
15. Isotomic Reciprocity
16. An Affine Property of Barycenter
17. Problem in Direct Similarity
18. Circles in Barycentric Coordinates
19. Barycenter of Cevian Triangle
20. Concurrent Chords in a Circle, Equally Inclined

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 Cradle
• Cevian Cradle II
• 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
• Menelaus from 3D
• Terquem's Theorem
• Cross Points in a Polygon
• Two Cevians and Proportions in a Triangle, II
• Concurrence Not from School Geometry
• Two Triangles Inscribed in a Conic - with Elementary Solution
• From One Collinearity to Another
• Concurrence in Right Triangle
  

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