Menelaus's theorem
In Euclidean geometry, Menelaus's theorem, named for Menelaus of Alexandria, is a proposition about triangles in plane geometry.
Suppose we have a triangle △ABC, and a transversal line that crosses BC, AC, AB at points D, E, F respectively, with D, E, F distinct from A, B, C. A weak version of the theorem states that
{\displaystyle \left|{\frac {\overline {AF}}{\overline {FB}}}\right|\times \left|{\frac {\overline {BD}}{\overline {DC}}}\right|\times \left|{\frac {\overline {CE}}{\overline {EA}}}\right|=1,}
where "| |" denotes absolute value (i.e., all segment lengths are positive).
The theorem can be strengthened to a statement about signed lengths of segments, which provides some additional information about the relative order of collinear points.
Here, the length AB is taken to be positive or negative according to whether A is to the left or right of B in some fixed orientation of the line; for example,
The signed version of Menelaus's theorem states
{\displaystyle {\frac {\overline {AF}}{\overline {FB}}}\times {\frac {\overline {BD}}{\overline {DC}}}\times {\frac {\overline {CE}}{\overline {EA}}}=-1.}
{\displaystyle {\overline {AF}}\times {\overline {BD}}\times {\overline {CE}}=-{\overline {FB}}\times {\overline {DC}}\times {\overline {EA}}.}
{\displaystyle {\frac {\overline {FA}}{\overline {FB}}}\times {\frac {\overline {DB}}{\overline {DC}}}\times {\frac {\overline {EC}}{\overline {EA}}}=1,}
The converse is also true: If points D, E, F are chosen on BC, AC, AB respectively so that
{\displaystyle {\frac {\overline {AF}}{\overline {FB}}}\times {\frac {\overline {BD}}{\overline {DC}}}\times {\frac {\overline {CE}}{\overline {EA}}}=-1,}
(Note that the converse of the weaker, unsigned statement is not necessarily true.)
By re-writing each in terms of cross-ratios, the two theorems may be seen as projective duals.
[3] A proof given by John Wellesley Russell uses Pasch's axiom to consider cases where a line does or does not meet a triangle.
[4] First, the sign of the left-hand side will be negative since either all three of the ratios are negative, the case where the line DEF misses the triangle (see diagram), or one is negative and the other two are positive, the case where DEF crosses two sides of the triangle.
To check the magnitude, construct perpendiculars from A, B, C to the line DEF and let their lengths be a, b, c respectively.
{\displaystyle \left|{\frac {\overline {AF}}{\overline {FB}}}\right|=\left|{\frac {a}{b}}\right|,\quad \left|{\frac {\overline {BD}}{\overline {DC}}}\right|=\left|{\frac {b}{c}}\right|,\quad \left|{\frac {\overline {CE}}{\overline {EA}}}\right|=\left|{\frac {c}{a}}\right|.}
{\displaystyle \left|{\frac {\overline {AF}}{\overline {FB}}}\right|\times \left|{\frac {\overline {BD}}{\overline {DC}}}\right|\times \left|{\frac {\overline {CE}}{\overline {EA}}}\right|=\left|{\frac {a}{b}}\times {\frac {b}{c}}\times {\frac {c}{a}}\right|=1.}
For a simpler, if less symmetrical way to check the magnitude,[5] draw CK parallel to AB where DEF meets CK at K. Then by similar triangles
[6] Let D, E, F be given on the lines BC, AC, AB so that the equation holds.
The following proof[7] uses only notions of affine geometry, notably homotheties.
The composition of the three then is an element of the group of homothety-translations that fixes B, so it is a homothety with center B, possibly with ratio 1 (in which case it is the identity).
Therefore D, E, F are collinear if and only if this composition is the identity, which means that the magnitude of the product of the three ratios is 1:
{\displaystyle {\frac {\overrightarrow {DC}}{\overrightarrow {DB}}}\times {\frac {\overrightarrow {EA}}{\overrightarrow {EC}}}\times {\frac {\overrightarrow {FB}}{\overrightarrow {FA}}}=1,}
It is uncertain who actually discovered the theorem; however, the oldest extant exposition appears in Spherics by Menelaus.
[8] In Almagest, Ptolemy applies the theorem on a number of problems in spherical astronomy.
[9] During the Islamic Golden Age, Muslim scholars devoted a number of works that engaged in the study of Menelaus's theorem, which they referred to as "the proposition on the secants" (shakl al-qatta').
The complete quadrilateral was called the "figure of secants" in their terminology.
[9] Al-Biruni's work, The Keys of Astronomy, lists a number of those works, which can be classified into studies as part of commentaries on Ptolemy's Almagest as in the works of al-Nayrizi and al-Khazin where each demonstrated particular cases of Menelaus's theorem that led to the sine rule,[10] or works composed as independent treatises such as:
