Thales’ Theorem

There are several theorems that are attributed to Thales of Miletus, we are going to focus especially on two of them:

Thales’ Theorem

Thales’ Theorem is a special case of the inscribed angle theorem, it’s related to right triangles inscribed in a circumference.

Thales’ theorem states that if A, B, and C are distinct points on a circle with a center O (circumcenter) where the line AC is a diameter, the triangle Δ ABC has a right angle (90 ) in point B. Thus, Δ ABC is a right triangle.

In other words, the diameter of a circle always subtends a right angle to any point on the circle.

Proof of Thales’ Theorem

If we connect the circumcenter O to point B we create two triangles Δ ABO and Δ OBC that are both isosceles triangles because all radii r are equal (OA, OB and OC are equal). And, by the base angle theorem, their base angles are equal. Let’s label the base angles of Δ ABO ‘α’, and those of Δ ABO ‘β’.

Because they are isosceles triangles, they each have two equal angles: α and β (see figure above).

As in any triangle, the interior angles of the triangle Δ ABC add up to 180°:

Dividing the equality by 2:

Since α + β is the angle of Δ ABC at point B, Thales’ theorem is proved.

Intercept Theorem

The Intercept Theorem states that if two intersecting lines are cut by parallel lines, the line segments cut by the parallel lines from one of the lines are proportional to the corresponding line segments cut by them from the other line.

If any two lines (in the image: m and n) are cut by a series of parallel lines (in the image: r, s and t), the line segments that are formed in one of them are proportional to the corresponding line segments formed in the other line.

Applying the Intercept Theorem, it is true that:

Where r is the ratio.

The Intercept Theorem is related to similarity. In fact, is equivalent to the concept of similar triangles. Applying the intercept theorem to triangles, we can assert that if in a given triangle we draw a line parallel to one of its three sides, the new triangle generated will be similar to the first. That is, we will create two congruent triangles that will be in Thales position.

See picture above.

In a Δ ABC we draw a line segment A’C’, parallel to side AC. Appears a new Δ A’BC’ similar to the first one. They have three equal angles and their sides are proportionals.

According to the intercept theorem, it is true that:

This ratio holds between two sides of the same triangle and also between the corresponding sides of the other: