If a line intersects two sides of a triangle and is parallel to the third side, then it divides those two sides in the same ratio.
In other words, if a line intersects two sides of a triangle and is parallel to the third side, then the ratio of the lengths of the segments of the two sides that are intersected is equal to the ratio of the lengths of the other two sides of the triangle.
>Here's a diagram that illustrates Thales' Theorem:
```
A--------B
| |
| |
C--------D
If line EF is parallel to side AD, then:
AE / EC = BF / FD
```
[Proof]
We can prove Thales' Theorem using similar triangles.
First, we draw a line from A to D. This line intersects line EF at point G.
>Now, we have two triangles: ABC and ADG.
Triangle ABC is similar to triangle ADG because they have two equal angles: angle CAB is equal to angle DAG because they are alternate interior angles, and angle ABC is equal to angle ADG because they are corresponding angles.
Since triangles ABC and ADG are similar, then we have:
AB / AD = BC / DG
We also know that line EF is parallel to AD, so we have:
EF / DG = AB / AD
Combining these two equations, we get:
EF / DG = BC / DG
Simplifying this equation, we get:
EF = BC
Therefore, line EF divides sides AC and BD in the same ratio.