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
Contact Front page Contents Generalizations Geometry
Copyright © 19962018 Alexander Bogomolny