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 A_{1}A_{2}A_{3} be a triangle with points B_{1}, B_{2}, B_{3} in sidelines A_{2}A_{3}, A_{1}A_{3}, and A_{1}A_{2}, respectively. Menelaus' theorem holds when an even number (0 or 2) of points B are internal to the sides of ΔA_{1}A_{2}A_{3}; 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_{i}B_{i} are concurrent provided
 · 

According to Menelaus' theorem, the three points B_{i} are collinear provided
 · 

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,
Assume b_{1}, b_{2}, b_{3} are real numbers such that
 · 

so that b_{1}b_{2}b_{3} = 1 for Ceva's and b_{1}b_{2}b_{3} = 1 for Menelaus' theorems.
We now place additional three points C_{1}, C_{2}, C_{3} on the side lines A_{2}A_{3}, A_{1}A_{3}, and A_{1}A_{2}. The real numbers c_{1}, c_{2}, c_{3} play the role similar to that of b_{1}, b_{2}, b_{3}, with a change of direction, for example,
Theorem
C_{1}B_{2}, C_{2}B_{3} and C_{3}B_{1} are concurrent iff
(*)  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. 
Proof
For the proof we'll use the barycentric coordinates. For any point, P in the plane of ΔA_{1}A_{2}A_{3} the barycentric coordinates p_{1}:p_{2}:p_{3} could be found starting with the signed areas of triangles A_{1}A_{2}P, A_{2}A_{3}P and A_{3}A_{1}P:
p_{1} : p_{2} : p_{3} = [A_{1}A_{2}P] : [A_{2}A_{3}P] : [A_{3}A_{1}P].
where, say, the area [A_{1}A_{2}P] is positive if triangles A_{1}A_{2}P and A_{1}A_{2}A_{3} have the same orientation and is negative otherwise. In other words, [A_{1}A_{2}P] is positive iff P is on the same side of A_{1}A_{2} as A_{3}. The barycentric coordinates are homogeneous, meaning that, for any real k≠0,
p_{1} : p_{2} : p_{3} = kp_{1} : kp_{2} : kp_{3}.
In barycentric coordinates, B_{1}(0:b_{1}:1), B_{2}(1:0:b_{2}), B_{3}(b_{3}:1:0) and, correspondingly, C_{1}(0:c_{1}:1), C_{2}(1:0:c_{2}), C_{3}(c_{3}:1:0). Let X(x_{1}:x_{2}:x_{3}) be a generic variable point in the plane of ΔA_{1}A_{2}A_{3}. Then the lines B_{1}C_{3}, B_{2}C_{1}, and B_{3}C_{2} are given by the determinant equations
 = 0, 
 = 0, 
 = 0. 
These reduce, respectively, to
c_{3}x_{1} + x_{2}  b_{1}x_{3} = 0,
b_{2}x_{1}  c_{1}x_{2} + x_{3} = 0,
x_{1}  b_{3}x_{2}  c_{2}x_{3} = 0.
These three lines are concurrent iff their equations are linearly dependent which is only true when
 = 0, 
which is exactly 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.
Observe that if C_{1}, C_{2}, C_{3} coincide with A_{2}, A_{3}, A_{1}, respectively, then
I placed an interactive illustration of the theorems of Ceva and Menelaus on a separate page.
References
 M. S. Klamkin, A. Liu, Simultaneous Generalization of the Theorems of Ceva and Menelaus, Mathematics Magazine, Vol 65, No 1 (February 1992), pp. 4852
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
 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
 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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