Use of directed lengths I was studying Menelaus' Theorem which can be found here https://en.wikipedia.org/wiki/Menelaus%27s_theorem.
 I noticed they use directed lengths which specifies that points $E,D,F$(in reference to above link) are collinear. My doubt is when do we use this kind of notation, where the magnitude of a side is taken to be negative. 
 Can someone please provide clarity on this issue or recommend a source. Thanks a lot for any help :)
 A: Signed lengths are a way to reduce the number of variations of a diagram that can be drawn. Consider the following basic use case:

If points $A$, $B$, and $C$ are collinear, then $AB + BC = AC$.

This is definitely true when point $B$ is between point $A$ and point $C$. We introduce signed lengths to make this statement true no matter what order $A, B, C$ appear in.
Signed lengths can be used in any statement of affine geometry: statements that only use properties which are unaffected by affine transformations. For example, in affine geometry:

*

*we can talk about points being collinear, and lines being concurrent or parallel;

*we can talk about ratios of areas, or ratios of parallel line segments;

*we cannot compare line segments that are not parallel;

*we cannot talk about angles;

*we cannot even measure absolute lengths of line segments, only their ratios. (Statements like $AB + BC = AC$ are okay if they are understood as $\frac{AB}{AC} + \frac{BC}{AC}=1$, which is valid.)

In this setting, the ratio $\frac{AB}{CD}$ is positive if $AB$ and $CD$ point in the same direction (they are required to be parallel). We do not make statements about signed lengths that do not reduce to ratios in this way.
But we could define a signed length by picking a reference line segment $PQ$ in each class of parallel lines, which we treat as having length $+1$, and letting $AB$ have the signed length $\frac{AB}{PQ}$.

In general, any statement of affine geometry is automatically compatible with signed lengths. If we prove it for ordinary lengths when the points are in one configuration (with respect to order, points being inside a triangle, etc.) then we can deduce it for signed lengths and all configurations. For example, Ceva's theorem is often stated for ordinary lengths, with points $D, E, F$ lying on segments $BC, CA, AB$. But if we allow the points to lie anywhere on the lines $BC, CA, AB$, then it is true for signed lengths.
The handwavy reason for that is that theorems in affine geometry using signed lengths can be restated as polynomial equations in the coordinates of points. Checking a theorem for ordinary lengths in a large "region" of cases tells us that a polynomial equation holds in that region. But if a polynomial equation in $n$ variables holds over an open subset of $\mathbb R^n$, then it holds over all of $\mathbb R^n$, giving us the theorem for signed lengths. (Here, $n$ is the number of degrees of freedom in the diagram.)
