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 A_{1}A_{2}A_{3} be a triangle with points B_{1}, B_{2}, B_{3} and C_{1}, C_{2}, C_{3} in sidelines A_{2}A_{3}, A_{1}A_{3}, and A_{1}A_{2}, respectively. Assume b_{1}, b_{2}, b_{3} and c_{1}, c_{2}, c_{3} are real numbers such that
(*) | b_{1}b_{2}b_{3} + c_{1}c_{2}c_{3} + b_{1}c_{1} + b_{2}c_{2} + b_{3}c_{3} = 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 ΔA_{1}A_{2}A_{3} are divided into 3, 4, and 5 segments, there are twelve such concurrences whose existence is assured by the theorem.
What if applet does not run? |
References
- 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
- 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
- Theorems of Ceva and Menelaus, an Illustrated Generalization
- 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
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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