Theorem of Tales of Miletus: First and Second Theorem, Applications and Examples

The first and the second Theorem of Thales of Miletus they are based on determining triangles from other similar ones (first theorem) or circumferences (second theorem). They have been very useful in various areas; For example, the first theorem proved 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 of Thales stand out (in some texts they also write it as Thales) and their useful applications. These results have been used throughout history and have allowed solving a wide variety of geometric problems.

Tales of Miletus

First theorem of Tales

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

Before giving your statement, remember some notions of similarity of triangles. Essentially, two triangles are similar if their angles are congruent (they have the same measure). This gives rise to the fact that, if two triangles are similar, their corresponding sides (or homologs) are proportional.

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

In the previous figure, the triangles ABC and DEC are similar. The proportionality that is obtained due to this similarity also gives rise to a relation of proportionality between two sides of the same triangle and the two corresponding sides of the other. For example, taking into account the previous figure you would also have to: Another way in which you can see the first theorem of Thales, and that is also very useful, is this: if two lines L1 and L2 (any) are cut by parallel lines (any number of these), then the segments formed in L1 are proportional to the corresponding ones formed in L2.

You also get a relationship between the angles that are formed, as seen in the following figure.

Application of the first theorem of Tales

Among its multiple applications stands out one of particular interest and has to do with one of the ways in which measurements were made of large structures in antiquity, time in which Thales lived and in which the modern measuring devices were not available. 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 stuck a stick or cane vertically into the ground.

Then he 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, you can clear and know the height of the pyramid.

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

Examples of the first theorem of Tales

Find the value of x in each case:

First case

Solution

Here we have two lines cut by two parallel lines. By the first theorem of Thales one has that their respective sides are proportional. In particular:

Second case

Solution

Here we have two triangles, one of these formed by a segment parallel to one of the sides of the other (precisely the side of length x). By the first theorem of Tales you have to:

Second theorem of Tales

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

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

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

By way of justification, note that both OA and OB and OC correspond to the radius of the circumference; Therefore, their measurements are the same. From there it is obtained 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 you have to:

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 that 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.

Note also 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 the vertex B is completely determined by the half of 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.

Circumference circumscribed

Another way to see Thales' second theorem is through a circle circumscribed to a right triangle.

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

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

Application of the second theorem of Tales

A very important application of the second theorem of Thales, and perhaps the most used, is to find the lines tangent to a given circumference, by a point P external to this (known).

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

It is known that the segment that goes from the center of a circle to a point of tangency of it, is perpendicular to this tangent line. Then, the OTP angle is straight.

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

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

By the second theorem of Thales 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 then sufficient 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 (x-h) ² + (y-k) ² = c ² , where c is the radius and the point (h, k) is the center of the circle.

Knowing now the equations of both circumferences, we can intersect them by solving the system of equations formed by these, 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 straight lines passing through T and P, and by T 'and P.

Example of the second theorem of Tales

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 much does AB measure?

Solution

By the second theorem of Thales we have that the triangle ABC is a rectangle 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 to:

References

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