First Thales' Theorem
If any two lines are cut by several parallel lines, the segments determined on one of the lines are proportional to the corresponding segments on the other.
Examples
1 Lines 
and 
are parallel. Find the length of 
.
Solution:
Applying Thales' theorem, we have:
2 Lines are parallel. Can we confirm that is parallel to lines 
and 
?
Solution:
Yes, because Thales' theorem is satisfied, since:
Second Thales' Theorem
Given a triangle 
, if a segment 
is drawn parallel to one of the sides of the triangle, another triangle 
is obtained, whose sides are proportional to those of triangle .
Example
1 Find the measurements of segments and .
Applications of Thales' Theorem
Thales' theorem is used to divide a segment into several equal parts.
Example
Divide segment 
into 
equal parts.
1 Draw a ray originating from endpoint 
of the segment.
2 Taking any measurement as a unit, mark units of measurement on the ray starting from .
3 Through each of the divisions on the ray, draw lines parallel to the segment that connects 
with the last division on the ray. The points obtained on segment determine the equal parts into which it is divided.
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