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наука

научпоп

математика

образование

3Blue1Brown

wild mathing

геометрия

Савватеев

школа

11 класс

формула

теорема

доказательство

физика

GetAClass

эксперимент

теорема о трех силах

Торричелли

оптика

Vectozavr

00:00:01

[music]

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where to mark a point inside the depicted

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triangle so that the sum of the

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distance from it to the vertices is

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minimal we have already solved this problem

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in three ways,

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but today there will be 4 no less surprising

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imagine that the outgoing segments of the

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yellow node are not those that stretch to the

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vertices in the vertices themselves we will do

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holes through which we tie 3 weights of

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equal mass as in the figure, as a result

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the system will come into equilibrium with

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minimal potential energy, this

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means that the vertical threads

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will have the greatest length in total, then the sum of the

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lengths from the node to the vertices will be minimal

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and we have solved the famous Steiner problem,

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note that the forces acting to the knot

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are directed along the threads are equal in magnitude and

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in total they neutralize each other, the

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sum of any two vectors is equal to 3, all of

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this with a minus sign, this is

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only possible if the angle between any two

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vectors is 120 degrees,

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now another difficult creative

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task, two points are marked inside an acute angle

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where on the sides of the angle he will put

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two new points so that the length of the yellow

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polyline is minimal

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[music]

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let’s, as in the background problem, let’s

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mirror the fixed point

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relative to the sides of the angle, then we can

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shift two segments and then

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work with a new polyline having the

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same length, think now where how

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to put the points the

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sum of the lengths of the links of the broken line is not less than the

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length of the segment connecting the end of this

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broken line and how to achieve a minimum when

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the inequality turns into equality

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only when points b and c lie on the

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segment 1 d 1 the problem is solved but what does

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physics have to do with it and you think about a ray of light

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that comes from from one fixed

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point, reflected from each side of the angle,

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it must hit another point, according to the

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principle of the company, a ray of light will guide us along the

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shortest optical path, let's

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carefully look at how exactly the

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depicted angles are equal due to symmetry,

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but these two angles are vertical, which means the

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corners that I am now highlighting are also

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equal a similar story with the angles

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formed by the right ray, therefore

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points b and c need to be marked so that the

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angle of incidence is equal to the angle of reflection

00:02:43

[music]

00:02:50

dual, remember the theorem about three forces, with its

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help you can very beautifully prove

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the fact and those related to the competitiveness of three

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straight lines, for example, you know well that

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the bisectors of a triangle intersect at

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one point, how can this be explained from the point of

00:03:05

view of physics, imagine that a

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triangle is an absolutely rigid body

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that a swan, a cancer, and a pike on a plane are

00:03:12

trying in vain to move from their place and you need to

00:03:15

select three vectors directed along the

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bisectors so that the sum is equal to

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zero, but how can you guess this in advance for

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any triangle everything is very simple,

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first we will let pairs of vectors from the

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vertices of the triangle

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if you want the resulting force of

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each of the three pairs to be directed along the

00:03:33

bisector, then it is logical to choose the forces

00:03:36

equal in magnitude for each pair, but in

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addition it is desirable that these two

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forces are opposite in direction

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neutralized each other, which means their

00:03:46

modules should also be equal for other

00:03:49

pairs; the same thing means, ultimately,

00:03:52

we take 6 equal in module and now we

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add each pair of vectors according to the parallelogram rule

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and get three cherished vectors on the

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bisectors, but now we know for sure

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what these three non-parallel vectors

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act on an absolutely rigid body

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which, as a result, is in

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equilibrium;

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therefore, do not their actions

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intersect at one point, which is what was

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required to be proven

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[music]

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finally, the modest charm of the mass method,

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which has already become a favorite of many,

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try first to prove

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on your own in any way that the

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medians of a triangle intersect at

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one point and together we will try

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to place unit masses at the vertices, the

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lower two can be regrouped in the

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middle of the segment, not forgetting that

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the mass will be doubled and then we will find the

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center of mass of the entire system using the

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rule of leverage, note that. with mass

00:04:56

3 divides the median in the ratio of two to one,

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counting from the top, now we will find the center of

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mass in a different way, first

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we regroup these two points, then the

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remaining two, the center of mass is on

00:05:10

another media, but it also divides it

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in the ratio of two to one, it turns out that the

00:05:16

center of mass lies both on one segment

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and on the other, it means being the

00:05:21

only one located at the point of

00:05:23

intersection of the segments and in a similar

00:05:25

way ends up on the third media, not

00:05:28

that is, we not only proved that the

00:05:29

median of the triangle is competitive but also

00:05:32

found out that the point of intersection

00:05:34

they are divided in a ratio of two to one

00:05:36

and from the vertex if you liked this

00:05:39

topic, write in the comments everyone who was a

00:05:42

sponsor but youtube rather go

00:05:44

to boost thanks for the support think and

00:05:48

do critical math

00:05:50

happy

00:05:51

[music]

00:06:06

[music]

Description:

Может ли классическая механика и законы оптики выручить в сложных геометрических задачах? Трудно представить, но бывает и такое! Поддержите канал Wild Mathing! https://boosty.to/wildmathing Мои курсы: https://vk.com/market-135395111 VK: https://vk.com/wildmathing Задачник: https://vk.com/wall-135395111_14984 ЛИТЕРАТУРА Метод масс: https://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=mo&paperid=15&option_lang=rus Точка Торричелли: https://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=kvant&paperid=1769&option_lang=rus Статья по теме (С.М. Крачковский): https://mccme.ru/free-books/uchim/book-8.pdf ТРИ ДРУГИХ РЕШЕНИЯ первой задачи из нынешнего видео: https://www.youtube.com/watch?v=_lOH0r0i_Mc СОДЕРЖАНИЕ 0:00 — Точка Торричелли и потенциальная энергия 1:12 — Минимум в геометрии и принцип Ферма 2:51 — Инцентр и теорема о трех силах 4:25 — Метод масс и правило рычага 5:45 — Еще одна красивая теорема и титры БОЛЬШЕ ГЕОМЕТРИИ от Wild Mathing: 1. Удивительные факты с анимацией: https://www.youtube.com/watch?v=UlfNYVFi37U 2. Теоремы XX века: https://www.youtube.com/watch?v=PH7IDlYD7f8 3. Принцип Дирихле в геометрии: https://www.youtube.com/watch?v=PzYFHbsNuKM 4. Гармония четырехугольников (feat. МО): https://www.youtube.com/watch?v=cJWnxrzR2D8

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