Simultaneous Generalization of the Theorems of Ceva and Menelaus

Ceva's and Menelaus' theorems are useful tools in establishing concurrency of lines and collinearity of points. Ceva's theorem is an easy consequence of Menelaus' and the opposite is also true. Although similar, the two theorems work in complementary circumstances. Let A₁A₂A₃ be a triangle with points B₁, B₂, B₃ in sidelines A₂A₃, A₁A₃, and A₁A₂, respectively. Menelaus' theorem holds when an even number (0 or 2) of points B are internal to the sides of ΔA₁A₂A₃; Ceva's theorem holds otherwise, when an odd number (1 or 3) of points B are internal to the sides of the triangle.

According to Ceva's theorem, the cevians A_iB_i are concurrent provided

A1B3 B3A2 · A2B1 B1A3

·

A3B2 B2A1  =  1

According to Menelaus' theorem, the three points B_i are collinear provided

A1B3 B3A2 · A2B1 B1A3

·

A3B2 B2A1  =  -1

The converse theorems are also true and are easily shown by contradiction to be equivalent to the direct statements. Note that all the segments involved are thought to be directed so that, for example, A₂B₁ = - B₁A₂.

Assume b₁, b₂, b₃ are real numbers such that A₂B₁ = A₂A₃ / (1 + b₁) so that B₁A₃ = b₁A₂A₃ / (1 + b₁), and b₂ and b₃ are defined similarly. For convenience, we allow b₁ = -1 in case B₁ is a point at infinity. In these notations,

A1B3 B3A2 · A2B1 B1A3

·

A3B2 B2A1  =  1 b1b2b3

so that b₁b₂b₃ = 1 for Ceva's and b₁b₂b₃ = -1 for Menelaus' theorems.

We now place additional three points C₁, C₂, C₃ on the side lines A₂A₃, A₁A₃, and A₁A₂. The real numbers c₁, c₂, c₃ play the role similar to that of b₁, b₂, b₃, with a change of direction, for example, C₁A₃ = A₂A₃ / (1 + c₁), etc. Then the following theorem holds.

Theorem

C₁B₂, C₂B₃ and C₃B₁ are concurrent iff

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

Proof

For the proof we'll use the barycentric coordinates. For any point, P in the plane of ΔA₁A₂A₃ the barycentric coordinates p₁:p₂:p₃ could be found starting with the signed areas of triangles A₁A₂P, A₂A₃P and A₃A₁P:

p₁ : p₂ : p₃ = [A₁A₂P] : [A₂A₃P] : [A₃A₁P].

where, say, the area [A₁A₂P] is positive if triangles A₁A₂P and A₁A₂A₃ have the same orientation and is negative otherwise. In other words, [A₁A₂P] is positive iff P is on the same side of A₁A₂ as A₃. The barycentric coordinates are homogeneous, meaning that, for any real k≠0,

p₁ : p₂ : p₃ = kp₁ : kp₂ : kp₃.

In barycentric coordinates, B₁(0:b₁:1), B₂(1:0:b₂), B₃(b₃:1:0) and, correspondingly, C₁(0:c₁:1), C₂(1:0:c₂), C₃(c₃:1:0). Let X(x₁:x₂:x₃) be a generic variable point in the plane of ΔA₁A₂A₃. Then the lines B₁C₃, B₂C₁, and B₃C₂ are given by the determinant equations

x1   x2   x3 0   b1   1 1   c3   0

= 0,

x1   x2   x3 1   0   b2 0   1   c1

= 0,

x1   x2   x3 b3   1   0 c2   0   1

= 0.

These reduce, respectively, to

-c₃x₁ + x₂ - b₁x₃ = 0,
-b₂x₁ - c₁x₂ + x₃ = 0,
x₁ - b₃x₂ - c₂x₃ = 0.

These three lines are concurrent iff their equations are linearly dependent which is only true when

c3   -1   b1 b2   c1   -1 -1   b3   c2

= 0,

which is exactly b₁b₂b₃ + c₁c₂c₃ + b₁c₁ + b₂c₂ + b₃c₃ = 1.

Observe that if C₁, C₂, C₃ coincide with A₂, A₃, A₁, respectively, then c₁ = c₂ = c₃ = 0 and (*) becomes b₁b₂b₃ = 1. This is Ceva's theorem. On the other hand, if C₁, C₂, C₃ coincide with B₁, B₂, B₃, respectively, then b₁c₁ = b₂c₂ = b₃c₃ = 1, so that (*) reduces to b₁b₂b₃ + 1/b₁b₂b₃ = -2 or b₁b₂b₃ = -1. This is Menelaus' theorem.

I placed an interactive illustration of the theorems of Ceva and Menelaus on a separate page.

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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