MILETUS TALES THEOREM: FIRST, SECOND AND EXAMPLES - MATHS - 2025

The first and second theorems of Thales of Miletus are based on determining triangles from similar ones (first theorem) or from circles (second theorem). They have been very useful in various areas. For example, the first theorem was very useful for measuring large structures when there were no sophisticated measuring instruments.

Thales of Miletus was a Greek mathematician who provided great contributions to geometry, of which these two theorems stand out (in some texts he is also written as Thales) and their useful applications. These results have been used throughout history and have made it possible to solve a wide variety of geometric problems.

Thales of Miletus

Thales' First Theorem

Thales' first theorem is a very useful tool that, among other things, allows the construction of a triangle similar to another, previously known. From here various versions of the theorem are derived that can be applied in multiple contexts.

Before giving your statement, let us recall some notions of similarity of triangles. Essentially, two triangles are similar if their angles are congruent (they have the same measure). This results in the fact that if two triangles are similar, their corresponding (or homologous) sides are proportional.

Thales' first theorem states that if a line is drawn parallel to any of its sides in a given triangle, the new triangle obtained will be similar to the initial triangle.

A relationship is also obtained between the angles that are formed, as shown in the following figure.

Application

Among its many applications, one of particular interest stands out and has to do with one of the ways in which measurements were made of large structures in Antiquity, a time in which Thales lived and in which there were no modern measuring devices that they exist now.

It is said that this is how Thales managed to measure the highest pyramid in Egypt, Cheops. For this, Thales supposed that the reflections of the solar rays touched the ground forming parallel lines. Under this assumption, he nailed a stick or cane vertically into the ground.

He then used the similarity of the two resulting triangles, one formed by the length of the shadow of the pyramid (which can be easily calculated) and the height of the pyramid (the unknown), and the other formed by the lengths of the shadow and the height of the rod (which can also be easily calculated).

Using the proportionality between these lengths, the height of the pyramid can be solved and known.

Although this measurement method can give a significant approximation error with respect to the accuracy of the height and depends on the parallelism of the solar rays (which in turn depends on a precise time), it must be recognized that it is a very ingenious idea and that it provided a good measurement alternative for the time.

Examples

Find the value of x in each case:

Thales' second theorem

The second theorem of Thales determines a right triangle inscribed in a circle at each point of the same.

A triangle inscribed to a circumference is a triangle whose vertices are on the circumference, thus remaining contained in it.

Specifically, Thales' second theorem states the following: given a circle with center O and diameter AC, each point B on the circumference (other than A and C) determines a right triangle ABC, with right angle

By way of justification, let us note that both OA and OB and OC correspond to the radius of the circle; therefore, their measurements are the same. From this it follows that the triangles OAB and OCB are isosceles, where

It is known that the sum of the angles of a triangle is equal to 180º. Using this with triangle ABC we have:

2b + 2a = 180º.

Equivalently, we have that b + a = 90º and b + a =

Note that the right triangle provided by Thales' second theorem is precisely the one whose hypotenuse is equal to the diameter of the circumference. Therefore, it is completely determined by the semicircle that contains the points of the triangle; in this case, the upper semicircle.

Let us also observe that in the right triangle obtained by means of Thales' second theorem, the hypotenuse is divided into two equal parts by OA and OC (the radius). In turn, this measure is equal to the segment OB (also the radius), which corresponds to the median of the triangle ABC by B.

In other words, the length of the median of the right triangle ABC corresponding to vertex B is completely determined by half the hypotenuse. Recall that the median of a triangle is the segment from one of the vertices to the midpoint of the opposite side; in this case, the BO segment.

Circumscribed girth

Another way of looking at Thales' second theorem is through a circumference circumscribed to a right triangle.

In general, a circumference circumscribed to a polygon consists of the circumference that passes through each of its vertices, whenever it is possible to draw it.

Using Thales' second theorem, given a right triangle, we can always construct a circumference circumscribed to it, with a radius equal to half the hypotenuse and a circumcenter (the center of the circumference) equal to the midpoint of the hypotenuse.

Application

A very important application of Thales' second theorem, and perhaps the most widely used, is to find the tangent lines to a given circle, through a point P external to it (known).

Note that given a circle (drawn in blue in the figure below) and an exterior point P, there are two lines tangent to the circle that pass through P. Let T and T 'be the points of tangency, r the radius of the circle, and Or the center.

It is known that the segment that goes from the center of a circle to a point of tangency of the same, is perpendicular to this tangent line. So the angle OTP is right.

From what we saw earlier in Thales' first theorem and its different versions, we see that it is possible to inscribe the OTP triangle in another circle (in red).

Similarly, it is obtained that the triangle OT'P can be inscribed within the same previous circumference.

By Thales' second theorem we also obtain that the diameter of this new circumference is precisely the hypotenuse of the triangle OTP (which is equal to the hypotenuse of the triangle OT'P), and the center is the midpoint of this hypotenuse.

To calculate the center of the new circumference, it is enough to calculate the midpoint between the center - say M - of the initial circumference (which we already know) and the point P (which we also know). Then the radius will be the distance between this point M and P.

With the radius and the center of the red circle we can find its Cartesian equation, which we remember is given by (xh) ² + (yk) ² = c ², where c is the radius and the point (h, k) is the center of the circumference.

Knowing now the equations of both circles, we can intersect them by solving the system of equations formed by them, and thus obtaining the points of tangency T and T '. Finally, to know the desired tangent lines, it is enough to find the equation of the lines that pass through T and P, and through T 'and P.

Example

Consider a circumference of diameter AC, center O, and radius 1 cm. Let B be a point on the circumference such that AB = AC. How tall is AB?

Solution

By Thales' second theorem we have that the triangle ABC is right and the hypotenuse corresponds to the diameter, which in this case measures 2 cm (the radius is 1 cm). Then, by the Pythagorean theorem we have:

References

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