Thales' theorem construction. Thales's theorem

About parallels and secants.

Outside the Russian-language literature, Thales' theorem is sometimes called another theorem of planimetry, namely, the statement that the inscribed angle subtended by the diameter of a circle is a right angle. The discovery of this theorem is indeed attributed to Thales, as evidenced by Proclus.

Formulations

If several equal segments are laid out in succession on one of two lines and parallel lines are drawn through their ends that intersect the second line, then they will cut off equal segments on the second line.

A more general formulation, also called proportional segment theorem

Parallel lines cut off proportional segments at secants:

Notes

• The theorem has no restrictions on mutual arrangement secants (this is true for both intersecting and parallel lines). It also does not matter where the segments on the secants are located.

• Thales's theorem is a special case of the proportional segments theorem, since equal segments can be considered proportional segments with a proportionality coefficient equal to 1.

Proof in the case of secants

Consider the option with unconnected pairs of segments: let the angle be intersected by straight lines A A 1 | | B B 1 | | C C 1 | | D D 1 (\displaystyle AA_(1)||BB_(1)||CC_(1)||DD_(1)) and wherein A B = C D (\displaystyle AB=CD).

Proof in the case of parallel lines

Let's make a direct B.C.. Angles ABC And BCD equal as internal crosswise lying with parallel lines AB And CD and secant B.C., and the angles ACB And CBD equal as internal crosswise lying with parallel lines A.C. And BD and secant B.C.. Then, by the second criterion for the equality of triangles, triangles ABC And DCB are equal. It follows that A.C. = BD And AB = CD. ■

Variations and generalizations

Converse theorem

If in Thales' theorem equal segments start from the vertex (often at school literature such a formulation is used), then the converse theorem will also be true. For intersecting secants it is formulated as follows:

In Thales' converse theorem, it is important that equal segments start from the vertex

Thus (see figure) from the fact that C B 1 C A 1 = B 1 B 2 A 1 A 2 = … (\displaystyle (\frac (CB_(1))(CA_(1)))=(\frac (B_(1)B_(2))(A_ (1)A_(2)))=\ldots ), follows that A 1 B 1 | | A 2 B 2 | | … (\displaystyle A_(1)B_(1)||A_(2)B_(2)||\ldots ).

If the secants are parallel, then it is necessary to require that the segments on both secants be equal to each other, otherwise this statement becomes false (a counterexample is a trapezoid intersected by a line passing through the midpoints of the bases).

This theorem is used in navigation: a collision between ships moving at a constant speed is inevitable if the direction from one ship to another is maintained.

Sollertinsky's lemma

The following statement is dual to Sollertinsky's lemma:

In the case of Thales's theorem, the conic will be the point at infinity, corresponding to the direction of parallel lines.

This statement, in turn, is a limiting case of the following statement:

This tomb is small, but the glory over it is immense.
The multi-intelligent Thales is hidden in it before you.

Inscription on the tomb of Thales of Miletus

Imagine this picture. 600 BC Egypt. Before you is a huge egyptian pyramid. To surprise the pharaoh and remain among his favorites, you need to measure the height of this pyramid. You have... nothing at your disposal. You can fall into despair, or you can act like Thales of Miletus: Use the triangle similarity theorem. Yes, it turns out that everything is quite simple. Thales of Miletus waited until the length of his shadow and his height coincided, and then, using the theorem on the similarity of triangles, he found the length of the shadow of the pyramid, which, accordingly, was equal to the shadow cast by the pyramid.

Who is this guy? Thales of Miletus? The man who gained fame as one of the “seven wise men” of antiquity? Thales of Miletus is an ancient Greek philosopher who distinguished himself with success in the field of astronomy, as well as mathematics and physics. The years of his life have been established only approximately: 625-645 BC

Among the evidence of Thales's knowledge of astronomy can be cited next example. May 28, 585 BC prediction by Miletus solar eclipse helped to end the war between Lydia and Media that had lasted for 6 years. This phenomenon frightened the Medes so much that they agreed to unfavorable terms for concluding peace with the Lydians.

There is a fairly widely known legend that characterizes Thales as a resourceful person. Thales often heard unflattering comments about his poverty. One day he decided to prove that philosophers can live in abundance if they wish. Even in winter, Thales determined from observing the stars that there would be a good harvest of olives in the summer. At the same time he hired oil presses in Miletus and Chios. This cost him quite little, since in winter there is practically no demand for them. When the olives produced a rich harvest, Thales began to rent out his oil presses. The large amount of money collected by this method was regarded as proof that philosophers can earn money with their minds, but their calling is higher than such earthly problems. This legend, by the way, was repeated by Aristotle himself.

As for geometry, many of his “discoveries” were borrowed from the Egyptians. And yet this transfer of knowledge to Greece is considered one of the main merits of Thales of Miletus.

The achievements of Thales are considered to be the formulation and proof of the following theorems:

• vertical angles are equal;
• Equal triangles are those whose side and two adjacent angles are respectively equal;
• the angles at the base of an isosceles triangle are equal;
• diameter divides the circle in half;
• the inscribed angle subtended by the diameter is a right angle.

Another theorem is named after Thales, which is useful in solving geometric problems. There is its generalized and particular form, the inverse theorem, the formulations may also differ slightly depending on the source, but the meaning of them all remains the same. Let's consider this theorem.

If parallel lines intersect the sides of an angle and cut off equal segments on one side, then they cut off equal segments on the other side.

Let's say points A 1, A 2, A 3 are the points of intersection of parallel lines with one side of the angle, and B 1, B 2, B 3 are the points of intersection of parallel lines with the other side of the angle. It is necessary to prove that if A 1 A 2 = A 2 A 3, then B 1 B 2 = B 2 B 3.

Through point B 2 we draw a line parallel to line A 1 A 2. Let's denote the new line C 1 C 2. Consider parallelograms A 1 C 1 B 2 A 2 and A 2 B 2 C 2 A 3 .

The properties of a parallelogram allow us to state that A1A2 = C 1 B 2 and A 2 A 3 = B 2 C 2. And since, according to our condition, A 1 A 2 = A 2 A 3, then C 1 B 2 = B 2 C 2.

And finally, consider the triangles Δ C 1 B 2 B 1 and Δ C 2 B 2 B 3 .

C 1 B 2 = B 2 C 2 (proven above).

This means that Δ C 1 B 2 B 1 and Δ C 2 B 2 B 3 will be equal according to the second sign of equality of triangles (by side and adjacent angles).

Thus, Thales' theorem is proven.

Using this theorem will greatly facilitate and speed up the solution of geometric problems. Good luck in mastering this entertaining science of mathematics!

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Plan:

Introduction

• 1 Converse theorem
• 2 Thales' theorem in culture
• 3 Interesting Facts
Notes

Introduction

This theorem is about parallel lines. For an angle based on a diameter, see another theorem.

Thales's theorem- one of the theorems of planimetry.

The theorem has no restrictions on the relative position of secants (it is true for both intersecting and parallel lines). It also doesn’t matter where the segments on the secants are.

Proof in the case of secants

Proof of Thales' Theorem

Consider the option with unconnected pairs of segments: let the angle be intersected by straight lines AA 1 | | BB 1 | | CC 1 | | DD 1 and wherein AB = CD .

Proof in the case of parallel lines

Let's draw a straight line BC. Angles ABC and BCD are equal as internal crosswise lying with parallel lines AB and CD and secant BC, and angles ACB and CBD are equal as internal crosswise lying with parallel lines AC and BD and secant BC. Then, by the first criterion for the equality of triangles, triangles ABC and DCB are congruent. It follows that AC = BD and AB = CD. ■

There is also generalized Thales' theorem:

Parallel lines cut off proportional segments at secants:

Thales' theorem is a special case of the generalized Thales' theorem, since equal segments can be considered proportional segments with a proportionality coefficient equal to 1.

1. Converse theorem

If in Thales’s theorem equal segments start from the vertex (this formulation is often used in school literature), then the converse theorem will also be true. For intersecting secants it is formulated as follows:

In Thales' converse theorem, it is important that equal segments start from the vertex

Thus (see figure) from what follows that straight lines .

If the secants are parallel, then it is necessary to require that the segments on both secants be equal to each other, otherwise this statement becomes false (a counterexample is a trapezoid intersected by a line passing through the midpoints of the bases).

2. Thales' theorem in culture

Argentine music group Les Luthiers ( Spanish) presented a song dedicated to the theorem. The video for this song provides a proof for the direct theorem for proportional segments.

3. Interesting facts

• Thales's theorem is still used in maritime navigation as a rule that a collision between ships moving at a constant speed is inevitable if the ships maintain a heading towards each other.
• Outside the Russian-language literature, Thales's theorem is sometimes called another theorem of planimetry, namely, the statement that the inscribed angle based on the diameter of a circle is right. The discovery of this theorem is indeed attributed to Thales, as evidenced by Proclus.
• Thales learned the basics of geometry in Egypt.

Notes

1. El Teorema de Thales por Les Luthiers en You Tube - www.youtube.com/watch?v=czzj2C4wdxY
2. 3. Travel to Egypt / Home / Ancient literature and philosophy. Thales from Miletus - www.fales-iz-mileta.narod.ru/3_puteshestvie_v_egipet

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This abstract is based on an article from Russian Wikipedia. Synchronization completed 07/16/11 23:06:34
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The theorem has no restrictions on the relative position of secants (it is true for both intersecting and parallel lines). It also does not matter where the segments on the secants are located.

Proof in the case of parallel lines

Let's draw a straight line BC. Angles ABC and BCD are equal as internal crosswise lying with parallel lines AB and CD and secant BC, and angles ACB and CBD are equal as internal crosswise lying with parallel lines AC and BD and secant BC. Then, according to the second criterion for the equality of triangles, triangles ABC and DCB are equal. It follows that AC = BD and AB = CD. ■

There is also proportional segment theorem:

Parallel lines cut off proportional segments at secants: $$

\frac(A_1A_2)(B_1B_2)=\frac(A_2A_3)(B_2B_3)=\frac(A_1A_3)(B_1B_3).

Thales's theorem is a special case of the proportional segments theorem, since equal segments can be considered proportional segments with a proportionality coefficient equal to 1.

Converse theorem

If in Thales’s theorem equal segments start from the vertex (this formulation is often used in school literature), then the converse theorem will also be true. For intersecting secants it is formulated as follows:

Thus (see figure) from the fact that $\backslash frac(CB\_1)(CA\_1)=\backslash frac(B\_1B\_2)(A\_1A\_2)=\backslash ldots\; =\; (\backslash rm\; idem)$ it follows that straight $A\_1B\_1||A\_2B\_2||\backslash ldots$.

If the secants are parallel, then it is necessary to require that the segments on both secants be equal to each other, otherwise this statement becomes false (a counterexample is a trapezoid intersected by a line passing through the midpoints of the bases).

Variations and generalizations

The following statement is dual to Sollertinsky's lemma:

Thales's theorem is still used in maritime navigation as a rule that a collision between ships moving at a constant speed is inevitable if the ships maintain a heading towards each other. Outside the Russian-language literature, Thales' theorem is sometimes called another theorem of planimetry, namely, the statement that the inscribed angle subtended by the diameter of a circle is a right angle. The discovery of this theorem is indeed attributed to Thales, as evidenced by Proclus. Write a review about the article "Thales' Theorem" Literature Atanasyan L. S. et al. Geometry 7-9. - Ed. 3rd. - M.: Education, 1992. Notes see also Thales' theorem on an angle subtended by the diameter of a circle Excerpt characterizing Thales' Theorem - I don’t think anything, I just don’t understand it... - Wait, Sonya, you will understand everything. You will see what kind of person he is. Don't think bad things about me or him. – I don’t think anything bad about anyone: I love everyone and feel sorry for everyone. But what should I do? Sonya did not give in to the gentle tone with which Natasha addressed her. The softer and more searching the expression on Natasha’s face was, the more serious and stern Sonya’s face was. “Natasha,” she said, “you asked me not to talk to you, I didn’t, now you started it yourself.” Natasha, I don't believe him. Why this secret? - Again, again! – Natasha interrupted. – Natasha, I’m afraid for you. - What to be afraid of? “I’m afraid that you will destroy yourself,” Sonya said decisively, herself frightened by what she said. Natasha's face again expressed anger. “And I will destroy, I will destroy, I will destroy myself as quickly as possible.” None of your business. It will feel bad not for you, but for me. Leave me, leave me. I hate you. - Natasha! – Sonya cried out in fear. - I hate it, I hate it! And you are my enemy forever! Natasha ran out of the room. Natasha no longer spoke to Sonya and avoided her. With the same expression of excited surprise and criminality, she walked around the rooms, taking up first this or that activity and immediately abandoning them. No matter how hard it was for Sonya, she kept an eye on her friend. On the eve of the day on which the count was supposed to return, Sonya noticed that Natasha had been sitting all morning at the living room window, as if expecting something, and that she made some kind of sign to a passing military man, whom Sonya mistook for Anatole. Sonya began to observe her friend even more carefully and noticed that Natasha was in a strange and unnatural state all the time during lunch and evening (she answered questions asked to her at random, started and did not finish sentences, laughed at everything). After tea, Sonya saw a timid girl's maid waiting for her at Natasha's door. She let her through and, listening at the door, learned that a letter had been delivered again. And suddenly it became clear to Sonya that Natasha had some terrible plan for this evening. Sonya knocked on her door. Natasha didn't let her in. “She'll run away with him! thought Sonya. She is capable of anything. Today there was something especially pitiful and determined in her face. She cried, saying goodbye to her uncle, Sonya recalled. Yes, it’s true, she’s running with him, but what should I do?” thought Sonya, now recalling those signs that clearly proved why Natasha had some terrible intention. “There is no count. What should I do, write to Kuragin, demanding an explanation from him? But who tells him to answer? Write to Pierre, as Prince Andrei asked, in case of an accident?... But maybe, in fact, she has already refused Bolkonsky (she sent a letter to Princess Marya yesterday). There’s no uncle!” It seemed terrible to Sonya to tell Marya Dmitrievna, who believed so much in Natasha. “But one way or another,” Sonya thought, standing in the dark corridor: now or never the time has come to prove that I remember the benefits of their family and love Nicolas. No, even if I don’t sleep for three nights, I won’t leave this corridor and forcefully let her in, and I won’t let shame fall on their family,” she thought. Anatole recently moved in with Dolokhov. The plan to kidnap Rostova had been thought out and prepared by Dolokhov for several days, and on the day when Sonya, having overheard Natasha at the door, decided to protect her, this plan had to be carried out. Natasha promised to go out to Kuragin’s back porch at ten o’clock in the evening. Kuragin had to put her in a prepared troika and take her 60 versts from Moscow to the village of Kamenka, where a disrobed priest was prepared who was supposed to marry them. In Kamenka, a setup was ready that was supposed to take them to the Warsaw road and there they were supposed to ride abroad on postal ones. Anatole had a passport, and a travel document, and ten thousand money taken from his sister, and ten thousand borrowed through Dolokhov. Two witnesses - Khvostikov, a former clerk, whom Dolokhov used for games, and Makarin, a retired hussar, a good-natured and weak man who had boundless love for Kuragin - were sitting in the first room having tea. In Dolokhov’s large office, decorated from walls to ceiling with Persian carpets, bear skins and weapons, Dolokhov sat in a traveling beshmet and boots in front of an open bureau on which lay abacus and stacks of money. Anatole, in an unbuttoned uniform, walked from the room where the witnesses were sitting, through the office into the back room, where his French footman and others were packing the last things. Dolokhov counted the money and wrote it down. “Well,” he said, “Khvostikov needs to be given two thousand.” “Well, give it to me,” said Anatole. – Makarka (that’s what they called Makarina), this one will selflessly go through fire and water for you. Well, the score is over,” said Dolokhov, showing him the note. - So? “Yes, of course, so,” said Anatole, apparently not listening to Dolokhov and with a smile that never left his face, looking ahead of him.

Theorem 6.6 (Thales' theorem).If parallel lines intersecting the sides of an angle cut off equal segments on one side, then they cut off equal segments on the other side.(Fig. 131).

Proof. Let A 1, A 2, A 3 be the intersection points of parallel lines with one of the sides of the angle and A 2 lies between A 1 and A 3 (Fig. 131). Let B 1, B 2, B 3 be the corresponding points of intersection of these lines with the other side of the angle. Let us prove that if A 1 A 2 = A 2 Az, then B 1 B 2 = B 2 B 3.

Let us draw a straight line EF through point B 2, parallel to straight line A 1 A 3. By the property of a parallelogram, A 1 A 2 = FB 2, A 2 A 3 = B 2 E. And since A 1 A 2 = A 2 A 3, then FB 2 = B 2 E.

Triangles B 2 B 1 F and B 2 B 3 E are equal according to the second criterion. They have B 2 F=B 2 E according to what has been proven. The angles at the vertex B 2 are equal as vertical, and the angles B 2 FB 1 and B 2 EB 3 are equal as internal crosswise lying with parallel A 1 B 1 and A 3 B 3 and the secant EF.

From the equality of triangles follows the equality of sides: B 1 B 2 = B 2 B 3. The theorem has been proven.

Comment. In the conditions of Thales's theorem, instead of the sides of an angle, you can take any two straight lines, and the conclusion of the theorem will be the same:

Parallel lines that intersect two given lines and cut off equal segments on one line also cut off equal segments on the other line.

Sometimes Thales' theorem will be applied in this form.

Problem (48). Divide this segment AB into n equal parts.

Solution. Let us draw from point A a half-line a that does not lie on line AB (Fig. 132). Let us plot equal segments on the half-line a: AA 1, A 1 A 2, A 2 A 3, .... A n - 1 A n. Let's connect points A n and B. Draw through points A 1, A 2, .... A n -1 lines parallel to straight line A n B. They intersect the segment AB at points B 1, B 2, B n-1, which divide the segment AB into n equal segments (according to Thales’ theorem).

A. V. Pogorelov, Geometry for grades 7-11, Textbook for educational institutions