I want to tell you
Dear friend:
Aphrodite is in front of you -
Stupid little girl!
Are you counting all your calories?
Do you always dream about compliments?
It's good to lose weight! It's good to fast!
It's time to eat and drink!
For sweet dreams you need a pillow,
For women's holidays - flowers.
And for happy days - a girlfriend,
So cool like YOU!!!
Oh, soul friend,
Main competitor
You are a good girl
How glorious!
You turn men's heads,
They like you very much
And you yourself love very much
Only Sochi resorts!
If someone didn't give in
to your strong charms,
Don't get caught again
It's a GIFT to the eye!
You are the secrets of charm
Don't tell anyone
Just one note here:
Tell me everything!
I'm your friend,
A curious little thing
You look with disbelief in vain,
I wouldn't spill the beans!
If only on the sly
In secret too...
But secrets from me
No and it cannot be!
And then with you alone
(We are maiden garns!)
Let's burn! Oh, we'll rock!
Watch out guys!
I have you! This is simply wonderful!
We will always support each other, we will understand!
May Life give you smiles and happiness,
Wonderful Tenderness, good luck in everything!
Let sunlight fate will illuminate
Joy meets you again and again,
And everything you want will happen sooner
And Hope and Love will be with you.
They don’t choose Russians, they don’t find them, and they don’t call from the street on the doorstep.
Fate determines, chance brings together, and this chance is sent by God!
A friend is not a second self, not a reflection, and sometimes strokes against the grain,
But in difficult times there is no need to call for achievements and hardships, he will come himself, ALWAYS!
I'll tell you, I won't keep silent,
I always want this.
I want to be in a field, even in a den,
I want it either in the tundra or in a den.
I want it everywhere, and wherever I have to.
Let the sun shine, let the rain pour.
I want when I go to bed
And in the morning, before getting up.
On the nightstand and on the sofa,
Squatting and upside down,
On land, in the air, in water.
I want to go to Moscow and Kostroma.
Both in the cold and in the summer heat...
I always want to chat with you.
You are so beautiful, like a princess!
I wish you a glorious process:
Find such a fellow
So that you can be brave with him even to the crown!
Try not to follow the principle -
He may not be a prince at all,
Well, rely on me -
I'll find you a white horse!
Sometimes you don't need a lot of words
There are times when they are meaningless
Stay close, giving your friend warmth
And, help with a look, only a sincere one.
And he can tell you a lot
And can replace many words,
And your friend will understand what you would like to say,
You will read tenderness and love in your eyes.
When we are close and shoulder to shoulder -
Our souls enter into conversation,
There is warmth from them,
It will also replace many words.
There are days when all words are unnecessary...
Save them all for later,
And only presence will not be superfluous,
To warm a friend with your warmth.
Or maybe, I’m not in the mood,
Just sit with a friend:
Don't talk, don't smile,
Just look into her eyes!…
She will understand, she will forgive,
If you feel bad,
Together we will share happiness,
Protect if necessary!
This happens very often
Replaces everyone sometimes
I know friendship is not in vain,
We stand by each other!
We were and remain goddesses,
Driven by the madness of their bodies.
Let those who didn’t get us lick their lips,
Let those who did not want us die!
Happy birthday pretty!
Let everything in life be clear:
If I laugh, then I'll drop,
And love - at first sight.
The wallet is big and thick,
And the eyes are like stars in the sky,
The whole world is at your disposal:
Always be positive!
Friend of my harsh days,
My decrepit dove,
You've aged another year
But she's still cute.
Rake your sand into a box
And hide it somewhere in the attic.
Open the sparkling cap,
Make a little bit of a mess.
On this day you are more beautiful than the sky.
Balloons, flowers - everything is for you.
And I know for sure: our friendship
It will pass through many summers.
Let you grow a year older, let your body hang and shake here and there, the main thing is that your soul sings all the time, and your bottom doesn’t sit still. I wish you a lot of gunpowder in your flask so that you can always set fire to the atmosphere and hearts of people. May the fire in your eyes, fire in your heart and passion in life never go out. Happy birthday, my aging, decrepit, but still beautiful and beloved friend!
May you, my friend,
Happiness smiles
And loved one for you
Day and night he tries:
May he please you in everything,
Let him carry it in his arms,
Coffee and toast in bed
Every day brings.
Did you dream? ! And now just -
Happy birthday!
I wish you from the bottom of my heart
Happiness and luck.
Peace, joy, goodness,
Good health.
And it doesn’t matter - coffee, tea,
The main thing is with love!
You always live like a fool,
Getting better every year.
Achieve all your desires
Don't give up on problems.
Neither fat nor thin,
But, of course, younger.
Have everything you like
Maserati in the garage.
To the resort - a pack of vouchers,
An endless stash
A fur coat, a cool watch,
Sandals are expensive.
Chanel and Prada handbag.
What else, girlfriend, do you need?
Groom, better,
And a bag of fans!
Happy Day, dear friend,
Be the same laugher
Stay young
And always by yourself!
A basket full of happiness,
Be as beautiful as a picture
Be loved so that you can love each other
And always live beautifully!
You're older today
And age another year.
You'll soon become very scary
And big, like a hippopotamus.
You'll eat off your belly and thighs,
And you start walking
Because the elevator can't
Make the climb with you.
Soon you'll be covered in wrinkles,
You'll rush to buy creams,
But cosmetics can't
He will hide all this.
But of course, you know
I love you any!
Happy birsday, my dear!
(This is humor like this:)
Happy birthday my friend.
Let the blizzard in life subside.
I always wish you
And the Bahamas and Hawaii,
In stores - everything is at a discount,
And work so that - with a smile.
May you always be loved
To be carried in your arms,
So that breakfasts are in bed,
At Disneyland there are carousels,
And the walks are somewhere in Nice.
No calories for pizza,
A bag - for every pair of shoes,
And for dinner - truffle.
So that the fans crowd
Followed you
So that there is plenty of everything,
And so that you love yourself!
Tender words and new clothes,
Travel to the seas
And fraying tights
I wish from the bottom of my heart.
To prevent weight gain,
The wallet was not empty
And admired with delight
Everyone who met you!
We've eaten a pound of salt over the years,
Darling, we are with you.
Tears were shed, songs were sung,
You have always been with me.
Happy birthday
And I hasten to tell you:
I'm with such a nice friend
It's easy to walk according to fate.
Be healthy and loved,
Find happiness in life.
May the sun always shine
In the island of your soul.
Well, friend, happy “Vareniya” Day,
I send you congratulations:
Be beautiful, be smart
Remember, you are given one life!
So, go ahead and “flutter”
On the go - don't fall asleep...
Sing, frolic, catch your luck,
With her you get more “comfort”!
Be as light as that feather
Be slim like that aspen tree
Be, dear, like a diamond,
The sparkle of the eyes, so as not to go out!
We are women, sisters, we love each other,
We cherish, we cherish, we are friends so honestly!
We do not close the spiritual circle,
We hear, we remember girlfriends everywhere.
We can congratulate, we can console,
We will not hurt you, we will support you in grief.
We will make you laugh, have fun,
We give our friends hope for happiness.
After all, a woman is wisdom, a mystery and a joke.
This is why I love my friends - terribly!
Let's grab a bottle of wine
Let the cigarettes freeze in your fingers
We'll sit in the kitchen by the window
We will reveal all our secrets to each other.
We will drink to love, to life, to our destiny,
My beloved and faithful friend,
Let those who didn’t get us cry,
Let those who did not want us die.
Come for us, come for our friendship,
Let her become stronger day by day,
I'm raising a toast to my best friend
Until the bottom I am for you, you are for me...
I feel good with you together
Talk about this and that.
Over a cup of tea and wine
And a delicious sweet pie
We'll wipe out all the secrets
At our kitchen table
Who lives how and “what about him?”
"How did everything go?" and then what?"
“Don’t think about bad things, Sun!
Do not be sad! Let's sing!"
Let's wave our hand and “soup with the cat!”
And let's go for a walk together,
And we will find adventures...
And on this bright summer day
I just want to say one thing:
That even after many years
Will you help with business, give advice,
We will find the answer to everything!
There is simply no better girlfriend!
Everything will be: joy and peace!
Love, shine, dance and sing!
And we will be happy with you,
My soulmate.
You will be by my side forever,
My dear, we are friends with you.
I'm very lucky, I'm not alone,
You were simply given to me from above!
Your eyes always shine for me
We are on the same wavelength day and night.
You will show me the right path in everything,
You can push away my trouble.
Give me a sea of golden ideas,
You will make an ordinary day bright.
You will always open the door for me,
And you won’t betray when the shadow overtakes you.
Two women were sitting by the window.
Girlfriends. We met in a cafe by chance.
Savoring red wine to the dregs,
They shared everything that happened, the past secret.
Two lives, like two poles of fate,
And between them there are a million paths.
The first life consisted of struggle,
And the second one has a continuous cascade of pictures.
One suffered, one endured pain,
And I’m happy that I survived and woke up.
Life was good: I raised the kids,
Love touched her with its wing.
The second in luxury both day and night,
She lived only for herself, her beloved, bright self.
But there is no happiness, and there is nothing to help her.
Bored, smoke, swallowing a cigarette.
Two lives, like heaven and hell,
We rushed by like a speck of dust from the universe,
But how to understand: what is emptiness, what is treasure?
Everyone has their own destiny and secret.
IN modern mechanical engineering a lot of elements and spare parts are used, which have both external and internal circles in their structure. The most a shining example can serve as bearing housings, motor parts, hub assemblies and much more. In their production, not only high-tech devices are used, but also knowledge from geometry, in particular information about the circles of a triangle. We will get acquainted with this knowledge in more detail below.
In contact with
Which circle is inscribed and which is circumscribed?
First of all, remember that a circle is an infinite set of points at equal distances from the center. If inside a polygon it is possible to construct a circle that has only one common intersection point with each side, then it will be called inscribed. A circumscribed circle (not a circle, it's different concepts) is called this locus points at which the constructed figure with a given polygon common points there will be only the vertices of the polygon. Let's get acquainted with these two concepts in more detail. clear example(See Figure 1.).
Figure 1. Inscribed and circumscribed circles of a triangle
The image shows two figures of large and small diameters, the centers of which are G and I. Circle greater value is called the described neighborhood Δ ABC, and the small one is called, on the contrary, inscribed in Δ ABC.
In order to describe the surroundings of a triangle, it is required draw a perpendicular line through the middle of each side(i.e. at an angle of 90°) is the point of intersection, it plays key role. It will be the center of the circumscribed circle. Before finding a circle, its center in a triangle, you need to construct for each angle, and then select the point of intersection of the lines. It, in turn, will be the center of the inscribed neighborhood, and its radius under any conditions will be perpendicular to any of the sides.
To the question: “How many inscribed circles can there be for a polygon with three?” Let us answer right away that a circle can be inscribed in any triangle, and only one. Because there is only one point of intersection of all bisectors and one point of intersection of perpendiculars emanating from the midpoints of the sides.
Property of the circle to which the vertices of a triangle belong
The circumscribed circle, which depends on the lengths of the sides at the base, has its own properties. Let us indicate the properties of the circumcircle:
In order to more clearly understand the principle of the circumscribed circle, we solve simple task. Let us assume that we are given a triangle Δ ABC, the sides of which are 10, 15 and 8.5 cm. The radius of the circumscribed circle around the triangle (FB) is 7.9 cm. Find the degree measure of each angle and through them the area of the triangle.
Figure 2. Finding the radius of a circle using the ratio of sides and sines of angles
Solution: based on the previously stated theorem of sines, we find the value of the sine of each angle separately. By condition, it is known that side AB is 10 cm. Let’s calculate the value of C:
Using the values of the Bradis table, we find out that the degree measure of angle C is 39°. Using the same method, we can find the remaining measures of angles:
How do we know that CAB = 33°, and ABC = 108°. Now, knowing the values of the sines of each of the angles and the radius, let’s find the area by substituting the found values:
Answer: The area of the triangle is 40.31 cm², and the angles are 33°, 108° and 39°, respectively.
Important! When solving problems of this kind, it will be useful to always have Bradis tables or a corresponding application on your smartphone, since the manual process can take a long time. long time. Also, to save more time, it is not necessary to construct all three midpoints of the perpendicular or three bisectors. Any third of them will always intersect at the point of intersection of the first two. And for an orthodox construction, the third is usually completed. Maybe this is wrong when it comes to the algorithm, but on the Unified State Exam or other exams it saves a lot of time.
Calculating the radius of an inscribed circle
All points of a circle are equally distant from its center at the same distance. The length of this segment (from and to) is called the radius. Depending on what kind of environment we have, there are two types - internal and external. Each of them is calculated using its own formula and is directly related to the calculation of parameters such as:
- square;
- degree measure of each angle;
- side lengths and perimeter.
Figure 3. Location of the inscribed circle inside the triangle
You can calculate the length of the distance from the center to the point of contact on either side in the following ways: h through the sides, sides and corners(for an isosceles triangle).
Using a semi-perimeter
A semiperimeter is half the sum of the lengths of all sides. This method is considered the most popular and universal, because no matter what type of triangle is given according to the condition, it is suitable for everyone. The calculation procedure is as follows:
If given "correct"
One of the small advantages of the "ideal" triangle is that inscribed and circumscribed circles have their center at the same point. This is convenient when constructing figures. However, in 80% of cases the answer is “ugly.” What is meant here is that very rarely the radius of the inscribed neighborhood will be whole, rather the opposite. For simplified calculation, use the formula for the radius of the inscribed circle in a triangle:
If the sides are the same length
One of the subtypes of tasks for the state. exams will be to find the radius of the inscribed circle of a triangle, two sides of which are equal to each other and the third is not. In this case, we recommend using this algorithm, which will significantly save time on searching for the diameter of the inscribed region. The radius of an inscribed circle in a triangle with equal “sides” is calculated by the formula:
We will demonstrate a more clear application of these formulas in the following problem. Let us have a triangle (Δ HJI), into which the neighborhood is inscribed at point K. The length of side HJ = 16 cm, JI = 9.5 cm and side HI is 19 cm (Figure 4). Find the radius of the inscribed neighborhood, knowing the sides.
Figure 4. Finding the value of the radius of the inscribed circle
Solution: to find the radius of the inscribed environment, we find the semi-perimeter:
From here, knowing the calculation mechanism, we find out the following value. To do this, you will need the lengths of each side (given according to the condition), as well as half the perimeter, it turns out:
It follows that the required radius is 3.63 cm. According to the condition, all sides are equal, then the required radius will be equal to:
Provided that the polygon is isosceles (for example, i = h = 10 cm, j = 8 cm), the diameter of the inner circle centered at point K will be equal to:
The problem may contain a triangle with an angle of 90°; in this case, there is no need to memorize the formula. The hypotenuse of the triangle will be equal to the diameter. It looks more clearly like this:
Important! If the task is to find the internal radius, we do not recommend performing calculations using the values of the sines and cosines of angles, the table value of which is not precisely known. If it is impossible to find out the length otherwise, do not try to “pull out” the value from under the root. In 40% of problems, the resulting value will be transcendental (i.e. infinite), and the commission may not count the answer (even if it is correct) due to its inaccuracy or irregular shape submissions. Special attention Pay attention to how the formula for the circumradius of a triangle can be modified depending on the proposed data. Such “blanks” allow you to “see” the scenario for solving a problem in advance and choose the most economical solution.
Inner circle radius and area
To calculate the area of a triangle inscribed in a circle, use only radius and side lengths of the polygon:
If the problem statement does not directly give the value of the radius, but only the area, then the indicated area formula is transformed into the following:
Let us consider the effect of the last formula on more specific example. Suppose that we are given a triangle into which the neighborhood is inscribed. The area of the neighborhood is 4π, and the sides are 4, 5 and 6 cm, respectively. Let's calculate the area of a given polygon by calculating the semi-perimeter.
Using the above algorithm, we calculate the area of the triangle through the radius of the inscribed circle:
Due to the fact that a circle can be inscribed in any triangle, the number of variations in finding the area increases significantly. Those. Finding the area of a triangle requires knowing the length of each side, as well as the value of the radius.
Triangle inscribed in a circle geometry grade 7
Right triangles inscribed in a circle
Conclusion
From these formulas you can be sure that the complexity of any problem using inscribed and circumscribed circles lies only in additional actions to find the required values. Problems of this type require only a thorough understanding of the essence of the formulas, as well as the rationality of their application. From the practice of solving, we note that in the future the center of the circumscribed circle will appear in further geometry topics, so it should not be started. IN otherwise the solution may be delayed using unnecessary moves and logical conclusions.