The great physicist of the early twentieth century, Ernest Rutherford, said: “All science is either physics or stamp collecting.” In the previous article, we were just convinced that physics and the collection of facts are at different poles in the development of models. And today we have our first excursion into the past.

Let's start with Thales of Miletus, because according to historians, it was he who laid the first brick in the foundation of the building of modern science.

A small digression regarding the actual name of the hero of our story today. I don’t know about you, but in Soviet school I personally was taught Thales’ theorem with the emphasis on the first syllable. In fact, the original Greek Θαλῆς has the letter Θ first, which in Greek is pronounced like the English th in think. This letter (fita) was present in its pure form in the Russian language until 1918, until it was shortened along with yatya and izhitsa. After the reform, it was replaced as the hand takes it - usually as F (in this case), but often as T (theory, thesis, theater). By the way, the above-mentioned Rutherford is actually Rutherford, which in Russian becomes, at the will of the translator, either Rutherford, or Rotherford, or Rutherford. With the second syllable Fa- forest In Russia we have no luck at all. Any Greek will tell you that it is pronounced “-fox” and not “-les” (in general, in the Greek language, as you know, five letters “i” fall on the unfortunate pair of “e” - one of which is epsilon), besides It is on this syllable that the stress falls.

With Miletus everything is much simpler; we managed to replace the next “i” with an “e”. This city was located on the coast of Asia Minor, maybe someone went to that distant Turkish coast to rest? According to legend, it was founded by immigrants (or refugees) from Crete, and named in honor of Militus, the son of Apollo and a Cretan beauty. In the times we are describing (six hundred years BC) it was the richest city of the Hellenistic world with numerous colonies - the most famous of which for us is the city of Feodosia (again, originally with fita). By this time, the Greeks had already built a real civilization - and one of the consequences of this was that individuals began to have time left over from the stupid struggle for existence.

Thales was one of these lucky ones - the son of rich parents. Wait a second - this is no coincidence. In that distant era, it was not easy to earn a living through science; you didn’t have not only Nobel Prizes, but even cushy places in numerous universities or research institutes. Almost exclusively people who had already previously solved all material problems for themselves became philosophers (and this was then a synonym for scientist). Why almost - because after a couple of centuries science began to be so respected (and this is largely due to Thales) that some money began to be paid for wisdom (for example, for the purposes of educating golden youth). Nevertheless, if you count all the ancient Greek philosophers, then there will be fewer of them than junior researchers in any of our provincial universities. But they were much more useful - wasn’t it because of the right motivation? After all, they were hungry for truth and knowledge, and not for the thirteenth salary.

Nevertheless, Thales apparently squandered his parents' money... on science, for which he went from Miletus to Egypt, like our Lomonosov from Kholmogory to Moscow. In any case, it is known that upon returning home they mocked him - they say, what is the use of your learning. Thales put his simple-minded critics in their place as follows.

Having calculated (in an incomprehensible way) from the stars that a large olive harvest was expected, he bought up all the presses in the area in advance, and then, renting them out at a speculative price, earned himself a fortune.

The power of a person (his intellect) is the ability to foresee (and sometimes create) the future, combining models known to him. Thales was truly famous for the solar eclipse he predicted, which even stopped the war between the Lydians and the Medes. The ancient Greek hot guys had to quickly make peace so that the gods would change their anger to mercy. Everything worked out with the gods, as you already guessed.

Knowledge of the so-called saros or draconic period (approximately every 18 years the moon and sun exactly repeat their locations) could help Thales predict a solar eclipse. The Chaldeans (Babylon) and the Egyptians knew about this. This is, in our terminology, a model of the causal phase - that is, it works, but it is not clear why. In general, from a pragmatic point of view, this doesn’t matter, even if you guess on coffee grounds (or, as was fashionable then, by studying the offal of domestic animals or consulting with oracles), as long as you get the required result.

So, as you can see, the man studied for good reason. However, there, in Egypt, he was taught by priests in temples after initiation rituals. And he, returning home, in line with his democratic worldview (and he was on friendly terms with the legendary legislator of Athens Solon) began to distribute sacred knowledge left and right to anyone. In the manner of Prometheus. And there was not a single Zeus in the vicinity to punish him for this.

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 Lately moved to 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.

(this 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 secants

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Proof of Thales' Theorem

Let's consider the option with unconnected pairs of segments: let the angle be intersected by straight lines texvc not found; See math/README for setup help.): AA_1||BB_1||CC_1||DD_1 and wherein Unable to parse expression (Executable file texvc not found; See math/README for setup help.): AB=CD .

1. Let's draw through the points Unable to parse expression (Executable file texvc not found; See math/README for setup help.): A And Unable to parse expression (Executable file texvc not found; See math/README for setup help.): C straight lines parallel to the other side of the angle. Unable to parse expression (Executable file texvc not found; See math/README - help with setup.): AB_2B_1A_1 And Unable to parse expression (Executable file texvc not found; See math/README - help with setup.): CD_2D_1C_1. According to the property of a parallelogram: Unable to parse expression (Executable file texvc not found; See math/README - help with setup.): AB_2=A_1B_1 And Unable to parse expression (Executable file texvc not found; See math/README - help with setup.): CD_2=C_1D_1 .
2. Triangles Unable to parse expression (Executable file texvc not found; See math/README for setup help.): \bigtriangleup ABB_2 And Unable to parse expression (Executable file texvc not found; See math/README for setup help.): \bigtriangleup CDD_2 are equal based on the second sign of equality of triangles
   

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: Unable to parse expression (Executable file texvc not found; See math/README for setup help.): \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:

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

Thus (see figure) from the fact that Unable to parse expression (Executable file texvc not found; See math/README - help with setup.): \frac(CB_1)(CA_1)=\frac(B_1B_2)(A_1A_2)=\ldots = (\rm idem) it follows that straight Unable to parse expression (Executable file texvc not found; See math/README - help with setup.): A_1B_1||A_2B_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).

Variations and generalizations

The following statement is dual to Sollertinsky's lemma:

Let Unable to parse expression (Executable file texvc not found; See math/README for setup help.): f- projective correspondence between points on a line Unable to parse expression (Executable file texvc not found; See math/README for setup help.): l and straight Unable to parse expression (Executable file texvc not found; See math/README for setup help.): m. Then the set of lines Unable to parse expression (Executable file texvc not found; See math/README for setup help.): Xf(X) will be the set of tangents to some conic section (possibly degenerate).

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:

Thales' theorem in culture

Argentine music group Les Luthiers presented a song dedicated to the theorem.

• 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 didn’t understand where all my fatigue had gone, and why I suddenly completely forgot the promise I made to myself a moment ago not to interfere in any, even the most incredible, incidents until tomorrow, or at least until I had at least a little rest. But, of course, this again triggered my insatiable curiosity, which I had not yet learned to pacify, even when there was a real need for it...
Therefore, trying, as far as my exhausted heart allowed, to “switch off” and not think about our failed, sad and difficult day, I immediately eagerly plunged into the “new and unknown”, anticipating some unusual and exciting adventure...
We smoothly “slowed down” right at the very entrance to the stunning “ice” world, when suddenly a man appeared from behind a sparkling blue tree... She was a very unusual girl - tall and slender, and very beautiful, she would have seemed quite young , almost if it weren’t for the eyes... They shone with calm, bright sadness, and were deep, like a well with the purest spring water... And in these wondrous eyes lurked such wisdom that Stella and I had not yet been able to comprehend for a long time ... Not at all surprised by our appearance, the stranger smiled warmly and quietly asked:
- What do you want, kids?
“We were just passing by and wanted to look at your beauty.” Sorry if I disturbed you...” I muttered, slightly embarrassed.
- Well, what are you talking about! Come inside, it will probably be more interesting there... - waving her hand into the depths, the stranger smiled again.
We instantly slipped past her inside the “palace”, unable to contain the curiosity rushing out, and already anticipating something very, very “interesting” in advance.
It was so stunning inside that Stella and I literally froze in a stupor, our mouths open like hungry one-day-old chicks, unable to utter a word...
There was no so-called “floor” in the palace... Everything there floated in the sparkling silver air, creating the impression of sparkling infinity. Some fantastic “seats”, similar to groups of sparkling dense clouds accumulated in groups, swaying smoothly, hung in the air, sometimes becoming denser, sometimes almost disappearing, as if attracting attention and inviting you to sit on them... Silvery “ice” flowers, shining and shimmering, they decorated everything around, striking with the variety of shapes and patterns of the finest, almost jewelry petals. And somewhere very high in the “ceiling”, blinding with sky-blue light, huge ice “icicles” of incredible beauty hung, turning this fabulous “cave” into a fantastic “ice world”, which seemed to have no end...
“Come on, my guests, grandfather will be incredibly glad to see you!” – the girl said warmly, gliding past us.
And then I finally understood why she seemed unusual to us - as the stranger moved, a sparkling “tail” of some special blue material was constantly trailing behind her, which sparkled and curled like tornadoes around her fragile figure, crumbling behind her. with silver pollen...
Before we had time to be surprised by this, we immediately saw a very tall, gray-haired old man, proudly sitting on a strange, very beautiful chair, as if thereby emphasizing his importance to those who did not understand. He watched our approach completely calmly, not at all surprised and not yet expressing any emotions other than a warm, friendly smile.
The white, silver-shimmering, flowing clothes of the old man merged with the same, completely white, long hair, making him look like a good spirit. And only the eyes, as mysterious as those of our beautiful stranger, shocked us with boundless patience, wisdom and depth, making us shudder from the infinity visible in them...
- Hello, guests! – the old man greeted affectionately. – What brought you to us?
- Hello to you, grandpa! – Stella greeted joyfully.
And then, for the first time in the entire time of our already quite long acquaintance, I was surprised to hear that she had finally addressed someone as “you”...
Stella had a very funny way of addressing everyone as “you,” as if emphasizing that all the people she met, whether an adult or a completely toddler, were her good old friends, and that for each of them she had her heart wide open. the soul is open... Which, of course, instantly and completely endeared even the most withdrawn and loneliest people to it, and only very callous souls did not find a way to it.
– Why is it so “cold” here? – immediately, out of habit, questions started pouring in. – I mean, why do you have such an “icy” color everywhere?
The girl looked at Stella in surprise.
“I never thought about it...” she said thoughtfully. – Probably because we had enough warmth for the rest of our lives? We were burned on Earth, you see...