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 http://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

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  2. Barycentric Coordinates
  3. Barycentric Coordinates: a Tool
  4. Barycentric Coordinates and Geometric Probability
  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

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