Menelaus theorem (also known as Menelaus' theorem, Menelaus's theorem, as well as theorem of Menelaus; attributed to Menelaus of Alexandria) is a theorem about triangles in plane geometry. Given points A, B, C that form triangle ABC, and points D, E, F that lie on lines BC, AC, AB, then the theorem states that D, E, F are collinear if and only if:
In this equation, AB etc. represent measurements of line segments that are allowed negative values. Fraction AF/FB can be defined as having positive value only when line DEF intersects side AB, and similarly for the other fractions.
Proof
This is one of many proofs for this theorem.
The sign of the left-hand side of the theorem's equation can be checked.
Line DEF can only intersect the sides of triangle ABC an even number of times. Hence there are an odd number of negative terms, and the total product is negative.
Next, the magnitude can be checked.
Construct line segments that connect perpendicularly line DEF with vertices A, B, and C. With DEF as the base, let the altitudes of A, B, and C be a, b, and c. Due to similar triangles, the absolute value of the left-hand side of the theorem's equation simplifies to:
Furrthermore, we can prove by contradiction that if the theorem's equation holds, then D, E, F must be collinear.
Let there be a new point F' on AB different from F, and define the measurements of AF, AF', and AB as n, n', and s.
Suppose that F' also satisfies the equation. Then the fractions have equal value:
which simplifies to n = n'. This proves that only one point on line AB can satisfy the equation. With D and E fixed, if F satisfies the equation, then it must be the point collinear with D and E. Due to reasons of symmetry, the same thing can be said about points D and E.
See also