Theorem of Thales

When a set of parallel lines intersects two transversal lines, r and s, the corresponding segments on these transversals are directly proportional.  $$ AB:CD = A'B':C'D' $$
example of corresponding points

For instance, if we have two segments, AB and CD, on transversal r, then there will be two corresponding segments, A'B' and C'D', on the other transversal s that maintain the same ratio. This means that AB is to CD as A'B' is to C'D'.

$$ AB:CD = A'B':C'D' $$

This equation expresses the direct proportionality between the two pairs of segments.

The inverse theorem of Thales is also valid.

If for every pair of segments AB and CD on transversal r there is a corresponding pair of segments A'B' and C'D' on transversal s such that the ratio AB:CD=A'B':C'D' holds true, then the set of intersecting lines must be parallel.

The Theorem of Thales and its inverse establish the necessary and sufficient conditions for two or more lines to be parallel based on the concept of direct proportionality.

The Theorem of Thales (or Thales' Correspondence) is a fundamental principle in plane geometry. The discovery of this theorem is traditionally attributed to Thales of Miletus, an ancient Greek mathematician and philosopher, although it is likely that the concept was known before his time.

A Practical Example

Consider a set of parallel lines intersecting two transversal lines, r and s, at various points.

example of corresponding points

In this scenario, the corresponding segments AB and CD on line r and the corresponding segments A'B' and C'D' on line s satisfy the following direct proportion:

$$ AB:CD = A'B':C'D' $$

In other words, segment AB is to segment CD as segment A'B' is to segment C'D'.

The segments on the transversals are directly proportional.

The Inverse Theorem of Thales

If two pairs of segments AB and CD on a line r and two pairs of segments A'B' and C'D' on a line s are directly proportional $$ AB:CD = A'B':C'D' $$ and the segments AA' and BB' are parallel, then the segments CC' and DD' are also parallel.

In other words, if I have two segments on one line and another two on another line, and the ratio of the lengths of the segments is the same for both lines, then if the lines connecting the ends of one pair of segments are parallel, the lines connecting the ends of the other pair will also be parallel.

Example

For example, consider the segments AB and CD on line r and the segments A'B' and C'D' on line s.

the sum of two segments on r corresponds to the sum of two segments on s

The ratio between the two lengths is the same:

$$ AB:CD = A'B':C'D' $$

Moreover, the segments connecting the ends of the first pair of segments are parallel:

$$ AA' \parallel BB' $$

According to the inverse theorem of Thales, if these conditions are met, then the segments connecting the ends of the second pair must also be parallel:

$$ CC' \parallel DD' $$

The Proof

Let's begin by assuming a set of parallel lines a, b, c, d.

$$ a \parallel b \parallel c \parallel d $$

Now, consider two transversal lines r and s intersecting this set of parallel lines.

example of corresponding points

 

We need to prove that for any two segments AB and CD on transversal r, there exist two segments A'B' and C'D' on transversal s that are directly proportional.

$$ AB:CD = A'B':C'D' $$

Let R denote the set of all segments on transversal r and S denote the set of segments on transversal s.

$$ R = \{ \ \text{segments on r} \ \} $$

$$ S = \{ \ \text{segments on s} \ \} $$

According to a fundamental principle of geometry, every point on a transversal line r that intersects a set of parallel lines corresponds exactly to a point on another transversal line s intersecting the same set of lines.

In other words, every point on transversal r is associated with exactly one point on transversal s.

For example, point A corresponds to point A', point B corresponds to point B', and so on.

example of corresponding points

If there is a one-to-one correspondence between the points on transversal lines r and s, then there is also a one-to-one correspondence between the segments of transversals r and s.

For example, segment AB corresponds to A'B', segment BC corresponds to B'C, and so on.

From this, we deduce the existence of a one-to-one correspondence between the elements of sets R and S, meaning that for every segment on transversal r there is one and only one segment on transversal s.

the one-to-one correspondence between segments of transversals r and s

Note: Essentially, the one-to-one correspondence of segments is based on the fact that every intersection point on one transversal has a direct corresponding point on the other transversal, due to the nature of the intersecting parallel lines.

According to the criterion of direct proportionality, two quantities are directly proportional if the following conditions are met:

  1. For every pair of congruent segments on transversal r, there is a pair of congruent segments on transversal s.
    example of congruence
  2. The sum of two segments AB+CD on transversal r corresponds to the sum of the corresponding segments A'B'+C'D' on transversal s.
    the sum of two segments on r corresponds to the sum of two segments on s

These two conditions are satisfied due to the one-to-one correspondence between the segments of transversals r and s.

Therefore, the corresponding segments on the transversals are directly proportional.

Observations

Some observations, corollaries, and notes on the theorem of Thales:

  • The Theorem of the Line Parallel to a Side of a Triangle
    If a line is parallel to one side of a triangle and intersects the other two sides, it divides these sides into two pairs of directly proportional segments, and vice versa. This is a corollary of the theorem of Thales.

    Example: Line r is parallel to side AB of the triangle and intersects the other two sides AC and BC.
    line s parallel to AB
    According to the Theorem of Thales, the ratio between the segments AD/DC and BE/EC is the same: $$ \overline{AD} : \overline{DC} = \overline{BE} : \overline{EC} $$ Therefore, the segments AD/DC and BE/EC are directly proportional.

  • The Bisector Theorem
    The bisector of an angle of a triangle divides the opposite side into two segments directly proportional to the other two sides of the triangle. $$ \overline{BD} : \overline{CD} = \overline{AB} : \overline{AC} $$ the bisector theorem of an internal angle of a triangle

And so on.

 
 

Please feel free to point out any errors or typos, or share suggestions to improve these notes. English isn't my first language, so if you notice any mistakes, let me know, and I'll be sure to fix them.

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