why is the basic proportionality theorem always true? The statement:

A line drawn parallel to one side of a triangle divides the other two sides in the same ratio.

Now there are several ways to prove this, and, prove its converse too. Hence we can convince ourselves that it is in fact true. But why? What is it about nature that makes this to be "true"?

If you imagine two objects moving in uniform motion from point $A$ along line $AB$ and $AC$ at different velocities, $x$ and $y$. You take some intercepts of time $t_1, t_2, t_3, etc$. Lets just say the points they make are labelled likewise as, $A_{xt_1}, A_{xt_2}, etc$ & $B_{yt_1}, B_{yt_2}, etc$...You'd always notice that $$A_{xt_1}B_{yt_1} \parallel A_{xt_2}B_{yt_2} \parallel A_{xt_3}B_{yt_3} ...$$
I've been hitting a dead end with this question: why is it so?!
 A: In many languages, this is known as Thales' Theorem. English Wikipedia calls it the Intercept Theorem. 
Let me first say that I disagree with the comments that this is proved using similar triangles. It is instead Thales' Theorem that is generally used to prove the facts about similar triangles.
The English Wikipedia page gives a proof of the theorem using area. But since area is a complex concept, it is probably preferable to give a proof that doesn't use area. 
The theorem is usually proved in the following steps:


*

*Let a number of parallel lines intersect two lines $l_1$ and $l_2$ that are secant to them. If the parallel lines mark off equal segments on $l_1$, then they also mark off equal segments on $l_2$. The proof of this step uses facts about parallelograms and the criteria for congruence of triangles.

*Given three parallel lines intersecting two lines $l_1$ and $l_2$, assume that they mark off segments on $l_1$ that are in a certain rational ratio. Then the segments marked off on $l_2$ are in the same ratio. This is proved by inserting additional parallel lines so that all the lines are equally spaced.

*Point 2 is extended to irrational ratios. This is done by the absurd: assume the ratios are different on $l_1$ and $l_2$. Then there must be some rational ratio that is larger than one and smaller than the other. (Any two distinct real numbers have some rational number between them. Geometrically, this corresponds to using Archimedes' Axiom.) Then you use point 2 to obtain a contradiction.
Unfortunately, I don't know a book in English where this proof is carried out. But a proof can be found in Geometría elemental by Pogorelov (Spanish translation from Russian). It seems likely that the proof might also be given in Kiselev's geometry book that has recently been translated into English, but I don't know for sure.
A: The basic idea is that
the two triangles are similar,
because their corresponding angles
are the same.
From this
(which,
of course,
is the essence of the proof),
the corresponding sides
are proportional.
Sometimes
believing that a theorem is true
just comes down to understanding the proof.
A: After considerable thought, this can be proven using the help of other truths. For e.g Law of Sines, i.e,
$$\frac{a}{\sin{A}} = \frac{b}{\sin{B}}= \frac{c}{\sin{C}}$$
This is easy to prove. One has to realize that sine is also a ratio. (Plus points if you realize how that ratio is formed. You can easily figure this out from the proof). 

But the essence of that law is saying that
$$\frac{\text{length of a side}}{\text{ratio of (length of altitude) from other vertex to (length of corresponding adjacent side)}} = \text{constant}$$
The denominator term depends on the bearing of the angle subtended. In other words, the ratio of side length to sine of opposite angles always bear a constant value. 
Parallel lines simply gives another way to express the sine ratio. Hence plugging that into the sine law result, we get the desired ratio.
Hence that is why...or so I believe.
