Tales'theorem

This is the basic theorem of similarities.

The Tales' theorem says: If two straight lines, not necessarily parallel, are cut by a system of parallel lines, then the resultant segments on one of the two lines are proportional to the respective segments obtained on the other line.

A figure to illustrate the above statement:

it is satisfied that $\dfrac{AB}{A'B'}=\dfrac{BC}{B'C'}.$

For example, given the following figure, decide whether the resultant segments are similar or not.

As we can see in the figure, the segment lengths are as follows: $(AB = 5, A'B' = 2)$; $(BC = 10, B'C' = 4)$. By Tales'theorem, we can say that the segments among straight lines are similar because the ratios are equal:

$$\dfrac{AB}{A'B'}=\dfrac{5}{2}=\dfrac{10}{4}=\dfrac{BC}{B'C'}$$

Finally, we give an application that can be useful for solving certain exercises. Thanks to Tales' Theorem, we can calculate the height of an object, for example, a tree, by the following mechanism.

1. Let $C$ be the length of the shadow of the tree at a certain time.
2. Let $B$ be the length of the shadow of a small object, for example a pencil, at the same instant.
3. Let $A$ be the height of the pencil.

Then, it is satisfied that the height of the tree, called $H$, is given by the following equation: $$H=C\cdot\Big(\dfrac{A}{B}\Big)$$