It's not difficult. There is a simple expression to calculate them that is easy to remember. For example, if the coordinates of the ends of a segment are respectively equal to (x1; y1) and (x2; y2), respectively, then the coordinates of its middle are calculated as the arithmetic mean of these coordinates, that is:
That's the whole difficulty.
Let's look at calculating the coordinates of the center of one of the segments using a specific example, as you asked.
Task.
Find the coordinates of a certain point M if it is the middle (center) of the segment KR, the ends of which have the following coordinates: (-3; 7) and (13; 21), respectively.
Solution.
We use the formula discussed above:
Answer. M (5; 14).
Using this formula, you can also find not only the coordinates of the middle of a segment, but also its ends. Let's look at an example.
Task.
The coordinates of two points (7; 19) and (8; 27) are given. Find the coordinates of one of the ends of the segment if the previous two points are its end and middle.
Solution.
Let us denote the ends of the segment as K and P, and its middle as S. Let us rewrite the formula taking into account the new names:
Let's substitute known coordinates and calculate the individual coordinates:
After painstaking work, I suddenly noticed that the size of web pages is quite large, and if things continue like this, then I can quietly go wild =) Therefore, I bring to your attention a short essay dedicated to a very common geometric problem - about dividing a segment in this respect, And How special case, about dividing a segment in half.
For one reason or another, this task did not fit into other lessons, but now there is a great opportunity to consider it in detail and leisurely. The good news is that we'll take a break from vectors and focus on points and segments.
Formulas for dividing a segment in this regardThe concept of dividing a segment in this regard
The concept of dividing a segment in this regard
Often you don’t have to wait for what’s promised at all; let’s immediately look at a couple of points and, obviously, the incredible – the segment:
The problem under consideration is valid both for segments of the plane and for segments of space. That is, the demonstration segment can be placed as desired on a plane or in space. For ease of explanation, I drew it horizontally.
What are we going to do with this segment? This time to cut. Someone is cutting a budget, someone is cutting a spouse, someone is cutting firewood, and we will start cutting the segment into two parts. The segment is divided into two parts using a certain point, which, of course, is located directly on it:
In this example, the point divides the segment in such a way that the segment is half as long as the segment. You can ALSO say that a point divides a segment in a ratio (“one to two”), counting from the vertex.
On dry mathematical language this fact is written as follows: , or more often in the form of the usual proportion: . The ratio of segments is usually denoted as Greek letter"lambda", in this case: .
It is easy to compose the proportion in a different order: - this notation means that the segment is twice as long as the segment, but this does not have any fundamental significance for solving problems. It can be like this, or it can be like that.
Of course, the segment can easily be divided in some other respect, and to reinforce the concept, the second example:
Here the following ratio is valid: . If we make the proportion the other way around, then we get: .
After we have figured out what it means to divide a segment in this respect, we move on to considering practical problems.
If two points of the plane are known, then the coordinates of the point that divides the segment in relation to are expressed by the formulas: 
Where did these formulas come from? I know analytical geometry These formulas are strictly derived using vectors (where would we be without them? =)). In addition, they are valid not only for the Cartesian coordinate system, but also for an arbitrary affine coordinate system (see lesson Linear (non) dependence of vectors. Basis of vectors). This is such a universal task.
Example 1
Find the coordinates of the point dividing the segment in the relation if the points are known 
Solution: In this problem. Using the formulas for dividing a segment in this relation, we find the point: 
Answer:
Pay attention to the calculation technique: first you need to separately calculate the numerator and the denominator separately. The result is often (but not always) a three- or four-story fraction. After this, we get rid of the multi-story structure of the fraction and carry out the final simplifications.
The task does not require drawing, but it is always useful to do it in draft form:
Indeed, the relation is satisfied, that is, the segment is three times shorter than the segment . If the proportion is not obvious, then the segments can always be stupidly measured with an ordinary ruler.
Equally valuable second solution: in it the countdown starts from a point and the following relation is fair: 
(in human words, the segment is three times longer than the segment). According to the formulas for dividing a segment in this respect: 
Answer:
Please note that in the formulas it is necessary to move the coordinates of the point to the first place, since the little thriller began with it.
It is also clear that the second method is more rational due to simpler calculations. But still, this problem is often solved in the “traditional” manner. For example, if according to the condition a segment is given, then it is assumed that you will make up a proportion; if a segment is given, then the proportion is “tacitly” implied.
And I gave the second method for the reason that often they try to deliberately confuse the conditions of the problem. That is why it is very important to carry out a rough drawing in order, firstly, to correctly analyze the condition, and, secondly, for verification purposes. It's a shame to make mistakes in such a simple task.
Example 2
Points given 
. Find:
a) a point dividing the segment in relation to ;
b) a point dividing the segment in relation to .
This is an example for independent decision. Complete solution and the answer at the end of the lesson.
Sometimes there are problems where one of the ends of the segment is unknown:
Example 3
The point belongs to the segment. It is known that a segment is twice as long as a segment. Find the point if 
.
Solution: From the condition it follows that the point divides the segment in the ratio , counting from the vertex, that is, the proportion is valid: . According to the formulas for dividing a segment in this respect: 
Now we do not know the coordinates of the point :, but this is not a particular problem, since they can be easily expressed from the above formulas. IN general view It doesn’t cost anything to express, it’s much easier to substitute specific numbers and carefully figure out the calculations: 
Answer:
To check, you can take the ends of the segment and, using the formulas in in direct order, make sure that the ratio actually results in a point . And, of course, of course, a drawing will not be superfluous. And in order to finally convince you of the benefits of a checkered notebook, a simple pencil and a ruler, I propose a tricky problem for you to solve on your own:
Example 4
Dot . The segment is one and a half times shorter than the segment. Find a point if the coordinates of the points are known 
.
The solution is at the end of the lesson. By the way, it is not the only one; if you follow a different path from the sample, it will not be a mistake, the main thing is that the answers match.
For spatial segments everything will be exactly the same, only one more coordinate will be added.
If two points in space are known, then the coordinates of the point that divides the segment in relation to are expressed by the formulas:
.
Example 5
Points are given. Find the coordinates of a point belonging to the segment if it is known that 
.
Solution: The condition implies the relation: 
. This example was taken from a real test, and its author allowed himself a little prank (in case someone stumbles) - it would have been more rational to write the proportion in the condition like this: 
.
According to the formulas for the coordinates of the midpoint of the segment:
Answer: 
3D drawings for inspection purposes are much more difficult to produce. However, you can always do schematic drawing, in order to understand at least the condition - which segments need to be correlated.
As for fractions in the answer, don’t be surprised, it’s a common thing. I’ve said it many times, but I’ll repeat it: higher mathematics It is customary to use ordinary regular and improper fractions. The answer is in the form 
will do, but the option with improper fractions is more standard.
Warm-up task for independent solution:
Example 6
Points are given. Find the coordinates of the point if it is known that it divides the segment in the ratio.
The solution and answer are at the end of the lesson. If it is difficult to navigate the proportions, make a schematic drawing.
In independent and tests The considered examples occur both on their own and as part of larger problems. In this sense, the problem of finding the center of gravity of a triangle is typical.
I don’t see much point in analyzing the type of task where one of the ends of the segment is unknown, since everything will be similar to the flat case, except that there are a little more calculations. Let’s remember our school years better:
Formulas for the coordinates of the midpoint of a segment
Even untrained readers can remember how to divide a segment in half. The problem of dividing a segment into two equal parts is a special case of dividing a segment in this respect. The two-handed saw works in the most democratic way, and each neighbor at the desk gets the same stick:
At this solemn hour the drums beat, welcoming the significant proportion. And general formulas 
miraculously transformed into something familiar and simple: 
A convenient point is the fact that the coordinates of the ends of the segment can be rearranged painlessly: 
In general formulas, such a luxurious room, as you understand, does not work. And here there is no particular need for it, so it’s a nice little thing.
For spatial case the obvious analogy is correct. If the ends of a segment are given, then the coordinates of its midpoint are expressed by the formulas:
Example 7
A parallelogram is defined by the coordinates of its vertices. Find the point of intersection of its diagonals.
Solution: Those who wish can complete the drawing. I especially recommend graffiti to those who have completely forgotten their school geometry course.
According to the well-known property, the diagonals of a parallelogram are divided in half by their point of intersection, so the problem can be solved in two ways.
Method one: Consider opposite vertices 
. Using the formulas for dividing a segment in half, we find the middle of the diagonal:
Very often in Problem C2 you need to work with points that bisect a segment. The coordinates of such points are easily calculated if the coordinates of the ends of the segment are known.
So, let the segment be defined by its ends - points A = (x a; y a; z a) and B = (x b; y b; z b). Then the coordinates of the middle of the segment - let’s denote it by point H - can be found using the formula:
In other words, the coordinates of the middle of a segment are the arithmetic mean of the coordinates of its ends.
· Task . The unit cube ABCDA 1 B 1 C 1 D 1 is placed in a coordinate system so that the x, y and z axes are directed along edges AB, AD and AA 1, respectively, and the origin coincides with point A. Point K is the middle of edge A 1 B 1 . Find the coordinates of this point.
Solution. Since point K is the middle of the segment A 1 B 1, its coordinates are equal to the arithmetic mean of the coordinates of the ends. Let's write down the coordinates of the ends: A 1 = (0; 0; 1) and B 1 = (1; 0; 1). Now let's find the coordinates of point K:
Answer: K = (0.5; 0; 1)
· Task . The unit cube ABCDA 1 B 1 C 1 D 1 is placed in a coordinate system so that the x, y and z axes are directed along the edges AB, AD and AA 1, respectively, and the origin coincides with point A. Find the coordinates of the point L at which they intersect diagonals of the square A 1 B 1 C 1 D 1 .
Solution. From the planimetry course we know that the point of intersection of the diagonals of a square is equidistant from all its vertices. In particular, A 1 L = C 1 L, i.e. point L is the middle of the segment A 1 C 1. But A 1 = (0; 0; 1), C 1 = (1; 1; 1), so we have:
Answer: L = (0.5; 0.5; 1)
The simplest problems of analytical geometry.
Actions with vectors in coordinates
It is highly advisable to learn how to solve the tasks that will be considered fully automatically, and the formulas memorize, you don’t even have to remember it on purpose, they will remember it themselves =) This is very important, since other problems of analytical geometry are based on the simplest elementary examples, and it will be annoying to spend additional time eating pawns. There is no need to fasten the top buttons on your shirt; many things are familiar to you from school.
The presentation of the material will follow a parallel course - both for the plane and for space. For the reason that all the formulas... you will see for yourself.
Initial geometric information
The concept of a segment, like the concept of a point, line, ray and angle, refers to initial geometric information. The study of geometry begins with the above concepts.
Under " initial information"usually understand something elementary and simple. In understanding, perhaps this is so. However, such simple concepts are often encountered and turn out to be necessary not only in our Everyday life, but also in production, construction and other areas of our life.
Let's start with definitions.
Definition 1
A segment is a part of a line bounded by two points (ends).
If the ends of the segment are the points $A$ and $B$, then the resulting segment is written as $AB$ or $BA$. Such a segment contains the points $A$ and $B$, as well as all points on the line lying between these points.
Definition 2
The midpoint of a segment is the point on a segment that divides it in half into two equal segments.
If this is point $C$, then $AC=CB$.
The measurement of a segment occurs by comparison with a specific segment taken as a unit of measurement. The most commonly used is a centimeter. If in a given segment a centimeter is placed exactly four times, this means that the length of this segment is $4$ cm.
Let's introduce a simple observation. If a point divides a segment into two segments, then the length of the entire segment is equal to the sum of the lengths of these segments.
Formula for finding the coordinates of the midpoint of a segment
The formula for finding the coordinate of the midpoint of a segment applies to the course of analytical geometry on a plane.
Let's define coordinates.
Definition 3
Coordinates are specific (or ordered) numbers that show the position of a point on a plane, on a surface, or in space.
In our case, the coordinates are marked on a plane defined by the coordinate axes.
Figure 3. Coordinate plane. Author24 - online exchange of student work
Let's describe the drawing. A point is selected on the plane, called the origin. It is denoted by the letter $O$. Two straight lines (coordinate axes) are drawn through the origin of coordinates, intersecting at right angles, and one of them is strictly horizontal, and the other is vertical. This situation is considered normal. The horizontal line is called the abscissa axis and is designated $OX$, the vertical line is called the ordinate axis $OY$.
Thus, the axes define the $XOY$ plane.
The coordinates of points in such a system are determined by two numbers.
There are different formulas (equations) that determine certain coordinates. Typically, in an analytical geometry course, they study various formulas for straight lines, angles, the length of a segment, and others.
Let's go straight to the formula for the coordinates of the middle of the segment.
Definition 4
If the coordinates of the point $E(x,y)$ are the middle of the segment $M_1M_2$, then:
Figure 4. Formula for finding the coordinates of the middle of a segment. Author24 - online exchange of student work
Practical part
Examples from a school geometry course are quite simple. Let's look at a few basic ones.
For a better understanding, let’s first consider an elementary visual example.
Example 1
We have a picture:
In the figure, the segments $AC, CD, DE, EB$ are equal.
- The midpoint of which segments is point $D$?
- Which point is the midpoint of segment $DB$?
- point $D$ is the midpoint of segments $AB$ and $CE$;
- point $E$.
Let's look at another simple example in which we need to calculate the length.
Example 2
Point $B$ is the middle of segment $AC$. $AB = 9$ cm. What is the length of $AC$?
Since t. $B$ divides $AC$ in half, then $AB = BC= 9$ cm. Hence, $AC = 9+9=18$ cm.
Answer: 18 cm.
Other similar examples are usually identical and focus on the ability to compare length values and their representation with algebraic operations. Often in problems there are cases when the centimeter does not fit exactly the number of times into a segment. Then the unit of measurement is divided into equal parts. In our case, a centimeter is divided into 10 millimeters. Separately measure the remainder, comparing it with a millimeter. Let us give an example demonstrating such a case.