What is the center of a circle. How can you find the center of a circle using various methods. Why is understanding the center of a circle important in geometry and real-world applications.
The Fundamental Concept of a Circle’s Center
A circle is a geometrical shape that has fascinated mathematicians and artists alike for centuries. At its core, quite literally, lies the center of the circle. But what exactly is this center, and why is it so crucial to our understanding of circles?
The center of a circle is a fixed point from which all points on the circle’s circumference are equidistant. This single point serves as the origin for all radii and diameters of the circle. Understanding the center is fundamental to grasping the properties and behavior of circles in various mathematical and real-world contexts.
Key Properties of the Circle’s Center
- Equidistance: All points on the circle’s circumference are at an equal distance from the center.
- Intersection point: The center is where all radii and diameters intersect.
- Symmetry: The center acts as a point of symmetry for the entire circle.
- Reference point: Many circle-related calculations and constructions use the center as a reference.
Methods for Locating the Center of a Circle
Finding the center of a circle is a crucial skill in geometry, engineering, and various practical applications. There are several methods to accomplish this task, each suited to different scenarios and available information.
1. The Chord Method
This method involves using two chords to locate the center:
- Draw two chords (AB and CD) of equal length in the circle.
- Find the perpendicular bisector of each chord.
- The point where these bisectors intersect is the center of the circle.
Why does this work? The perpendicular bisector of a chord always passes through the center of the circle. By using two chords, we create two lines that must intersect at the center.
2. The Perpendicular Tangent Method
If you have access to the circle’s tangent lines:
- Draw two tangent lines to the circle at different points.
- Construct perpendicular lines from the points of tangency.
- The intersection of these perpendicular lines is the center of the circle.
This method works because the radius is always perpendicular to the tangent line at the point of tangency.
3. The Coordinate Geometry Method
When dealing with circles in a coordinate plane:
- Identify the equation of the circle in the form (x – h)² + (y – k)² = r².
- The coordinates (h, k) represent the center of the circle.
This method is particularly useful when working with analytical geometry or when the circle is defined by an equation rather than a physical drawing.
4. The Diameter Method
If you know the endpoints of a diameter:
- Identify the coordinates of the diameter’s endpoints: (x₁, y₁) and (x₂, y₂).
- Use the midpoint formula: Center = ((x₁ + x₂)/2, (y₁ + y₂)/2).
This method leverages the fact that the center of a circle is always the midpoint of any diameter.
The Mathematical Foundation of Circle Centers
The concept of a circle’s center is deeply rooted in mathematical principles. Understanding these foundations can provide insights into why the center behaves as it does and how it relates to other aspects of circle geometry.
The Circle Equation
The standard form of a circle’s equation is:
(x – h)² + (y – k)² = r²
Where (h, k) represents the coordinates of the center, and r is the radius. This equation encapsulates the definition of a circle: all points (x, y) that satisfy this equation are at a distance r from the center (h, k).
Distance Formula and the Center
The distance formula in coordinate geometry is derived from the Pythagorean theorem:
d = √[(x₂ – x₁)² + (y₂ – y₁)²]
When applied to circles, this formula helps us understand why all points on the circumference are equidistant from the center. For any point (x, y) on the circle:
r = √[(x – h)² + (y – k)²]
This relationship is the foundation of the circle’s definition and properties.
Practical Applications of Circle Centers
The concept of a circle’s center extends far beyond abstract mathematics. It has numerous practical applications in various fields, demonstrating the real-world relevance of this geometrical concept.
Engineering and Construction
- Designing circular structures: Architects and engineers use the center point as a reference for creating domes, arches, and circular buildings.
- Constructing roundabouts: Traffic engineers determine the center point to design efficient and safe circular intersections.
- Manufacturing circular parts: Precision machining often requires accurate identification of a circle’s center for proper alignment and balance.
Navigation and GPS
- Triangulation: GPS systems use the principle of circle intersections to determine location, with each satellite signal forming a sphere (a 3D circle) with the receiver at the center.
- Radar systems: The center of a radar’s circular display represents the observer’s position, crucial for air traffic control and maritime navigation.
Sports and Recreation
- Field layouts: Sports like football and cricket use the center of circular boundaries to define playing areas and strategic positions.
- Dartboards: The bullseye, often considered the most challenging target, is at the center of the circular board.
Art and Design
- Mandala creation: Artists use the center point as a starting reference for creating intricate, symmetrical designs.
- Pottery: Ceramic artists center their clay on the potter’s wheel to ensure symmetry and balance in their creations.
Advanced Concepts Related to Circle Centers
As we delve deeper into the study of circles, several advanced concepts emerge that are intimately connected to the center. These concepts not only expand our understanding of circles but also reveal the center’s crucial role in more complex geometrical relationships.
Concentric Circles
Concentric circles are circles that share the same center but have different radii. This concept is fundamental in various applications:
- Target designs: Archery targets and dartboards use concentric circles to assign different point values.
- Planetary models: Early astronomers used concentric circles to model planetary orbits.
- Sound wave propagation: Concentric circles can represent the spread of sound waves from a point source.
Circle Inversion
Circle inversion is a geometric transformation that maps points inside a circle to points outside, and vice versa. The center of the circle plays a crucial role in this transformation:
- Definition: For a circle with center O and radius r, the inverse of point P is P’, where OP × OP’ = r².
- Properties: Lines not passing through the center invert to circles, while lines through the center invert to themselves.
- Applications: Circle inversion is used in complex analysis, conformal mapping, and even in designing antennas.
Power of a Point
The power of a point with respect to a circle is a measure of how far the point is from the circle. It’s defined as the product of the distances from the point to the intersections of any line through the point with the circle.
- For points outside the circle: Power = d² – r², where d is the distance from the point to the center.
- For points inside the circle: Power = r² – d².
- Applications: This concept is used in constructing tangent lines and solving problems involving intersecting circles.
Circles in Non-Euclidean Geometry
While we’ve primarily discussed circles in Euclidean geometry, the concept of circles and their centers takes on fascinating new dimensions in non-Euclidean geometries.
Spherical Geometry
On the surface of a sphere, “circles” can be defined in two ways:
- Great circles: These are the largest circles that can be drawn on a sphere, and their centers coincide with the center of the sphere itself.
- Small circles: These have centers that lie on a line connecting the center of the sphere to the circle’s center on the surface.
Applications of spherical geometry include navigation, astronomy, and geodesy.
Hyperbolic Geometry
In hyperbolic geometry, circles can have some counterintuitive properties:
- The ratio of a circle’s circumference to its diameter is always greater than π.
- As circles get larger, their curvature appears to decrease, approaching a straight line called a horocycle.
Understanding circles in non-Euclidean geometries has applications in relativity theory and certain branches of mathematics.
Technological Advancements and Circle Centers
Modern technology has both benefited from and contributed to our understanding of circles and their centers. Let’s explore some cutting-edge applications and how they relate to this fundamental geometric concept.
Computer Vision and Image Processing
In the field of computer vision, detecting circles and their centers is a crucial task:
- Hough Transform: This algorithm is used to detect circles in images, identifying their centers and radii.
- Object Recognition: Circular features and their centers are often key points in identifying and tracking objects in video streams.
- Medical Imaging: Detecting circular structures is vital in analyzing medical scans for diagnoses.
3D Printing and Manufacturing
Precise identification of circle centers is essential in advanced manufacturing processes:
- Calibration: 3D printers often use circular calibration patterns to ensure accuracy.
- Tool Path Generation: CNC machines rely on accurate center points to create circular cuts and features.
- Quality Control: Automated inspection systems use circle detection to verify the dimensions and positions of manufactured parts.
Augmented Reality (AR) and Virtual Reality (VR)
Circle detection and center identification play roles in AR and VR technologies:
- Marker-based AR: Many AR applications use circular markers as anchor points, with their centers serving as reference points for overlaying digital content.
- Eye-tracking: VR headsets often use algorithms to detect the circular iris and its center for precise eye movement tracking.
As technology continues to evolve, our understanding of circles and their centers remains fundamental, finding new applications in increasingly sophisticated systems and devices.
Center of Circle – Definition, Formula, Examples, Facts
What Is the Center of a Circle?
A circle is the set of all points (or locus of all points) in the plane that lie at a fixed distance from a fixed point. The fixed point is called the “center” of the circle. The fixed distance is called “radius.”
Center of a Circle: Definition
A circle is a two-dimensional shape defined by its center and radius. We can draw any circle if we know the center and radius. A circle can have an infinite number of radii. The center point is the midpoint where all radii intersect. It can also be defined as the midpoint of the diameter of the circle.
We can construct a circle of any radius using a compass. The distance between the pencil and the pointer of a compass is adjustable. Construct the center of a circle by keeping the pointer on any fixed point. Set the distance equal to the required radius. Draw a complete circle without changing this distance. The fixed point acts as the center of the circle.
{2}$
How to Find the Center of a Circle
To find the center of the circle, we’ll consider two cases:
- We need to locate the center of the given circle.
- We need to find the coordinates of the center of the circle using the equation of the circle.
When a circle is given
When we’re given a circle and we need to find its center, we can follow the steps listed below:
Step 1: Draw a chord AB in the circle and note down its length carefully .
Step 2: Draw another chord CD parallel to AB so that it is the same length as AB.
Step 3: Use a ruler to connect points P and N with a line segment.
Step 4: Connect points B and C.
Step 5: The intersection of AD and BC is the center of the circle.
In a similar way, the center of a circle can be found using secants (chords are the part of secant inside the circle). Also, we can use the perpendiculars of tangents at the point of contact to find the center.
{2}\;-\;4y + 4$
Comparing with the general equation we have h $= 0$, k $= 2$ and r $= 2$. Thus the center of the circle is (0, 2).
When endpoints of a diameter are given
If the endpoints of the diameter are given, then to find the coordinates of the center
point, we use the midpoint formula, since the center is the midpoint of the
diameter. The steps to find the center of the two points are as follows:
Step 1: Assume that the circle’s center is at these coordinates $(h,k)$.
Step 2: If $(h, k)$ is the midpoint coordinates of a line segment with
endpoints $(x_{1},\;y_{1})$and $(x_{2},\;y_{2})$, then by midpoint formula we can write
$(h, k) = (\frac{x_{1}+ x_{2}}{2},\frac{y_{1} + y_{2}}{2})$
Step 3: Simplify to get the coordinates of the center of the circle.
Let’s take an example of a circle where the endpoints of the diameters are $( \;-2,\;4)$ and $4(6, \;16)$.
Then, its center coordinates are:
$(h,k) = (\frac{-2+6}{2},\frac{4+16}{2})$
$(h,k)=(\frac{4}{2},\frac{20}{2})$
$(h,k)=(2,\;10)$
Therefore, the coordinates of the center of the circle whose endpoint is the diameter are (2,10).
Fun Facts about the Center of a Circle
- If a circle is divided into two equal parts, each part is called a semicircle. The diameter of a circle divides it into 2 semicircles.
- A sector of a circle is part of a circle in between two radii and an arc; there is a unique sector, formed by radii at right angles and is known as a quadrant.
- The perpendicular bisectors of any two chords of a circle always meet at the center of a circle.
Conclusion
The center of a circle is a point inside the circle, which is equidistant from all the points on the boundary of the circle. In this article, we learned about the center of a circle, its properties, and how to find the center of a given circle using different methods.
Solved Examples on Center of a Circle
1. Find the equation of a circle, given the coordinates of the center are (3, 1), and the radius of the circle is 5 units. Check if the origin lies inside the circle, on the circle, or outside of the circle.
{2} = 12$ is the equation of a circle.
Expert Maths Tutoring in the UK
A circle is defined as the locus of a moving point on a plane such that its distance from a fixed point on the plane remains constant or fixed. That fixed point is called the center of the circle. Let us learn more about the center of a circle in this article.
| 1. | Center of Circle Definition |
| 2. | Center of Circle Formula |
| 3. | How to Find Center of Circle? |
| 4. | FAQs on Center of Circle |
Center of Circle Definition
A circle is a 2D shape defined by its center and radius. We can draw any circle if we know the center of circle and its radius. A circle can have an infinite number of radii. The center of a circle is the midpoint where all the radii meet.
It can also be defined as the midpoint of the diameter of the circle. Observe the figure given below where O is the center of circle and OP is the radius.
Center of Circle Formula
The center of circle formula is also known as the general equation of a circle. In a circle, if the coordinates of the center are (h,k), r is the radius, and (x,y) is any point on the circle, then the center of circle formula is given below:
(x – h)2 + (y – k)2 = r2
This is also known as the center of the circle equation. We will be using this formula in the following sections to find the center of a circle or the equation of the circle.
How to Find the Center of Circle?
In order to find the center of the circle, we will use some simple steps. There are two cases that might come up when we could be asked to find the center of a circle:
- When a circle is given and we need to find its center.
- When an equation of a circle is given and we need to find the coordinates of its center.
When a Circle is Given
When a circle is given to us and we need to find its center point, then we can follow the steps listed below:
Step 1: Draw a chord PQ in a circle and carefully note its length (which is 4 inches in the figure below).
Step 2: Draw another chord MN parallel to PQ such that it should be of the same length as PQ.
Step 3: Join the points P and N through a line segment using a ruler.
Step 4: Join points Q and M.
Step 5: The point of intersection of PN and QM is the center of the circle.
When Equation of the Circle is Given
If we know the equation of a circle, and we need to find its center, then we will use the following steps. Let us understand this with the help of an example.
Example: Let us find the coordinates of the center of a circle with equation x2 + y2 – 4x – 6y – 87 = 0
Solution: The steps to find the coordinates of the center of a circle are listed below:
- Step 1: Write the given equation in the form of the general equation of a circle: (x – h)2 + (y – k)2 = r2, by adding or subtracting numbers on both sides.
We can write the given equation as x2 – 4x + y2 – 6y = 87. Add 4 to both sides of the equation to get a perfect square of x-2. So, we will get, x2 – 4x + 4 + y2 – 6y = 87 + 4.
⇒ (x – 2)2 + y2 – 6y = 91
Add 9 to both sides to get a perfect square of y – 3
⇒ (x – 2)2 + y2 – 6y + 9 = 91 + 9
⇒ (x – 2)2 + (y – 3)2 = 100
⇒ (x – 2)2 + (y – 3)2 = 102
This looks like the general equation of circle.
- Step 2: Compare this equation with the general equation and identify the values of h, k, and r.
If we compare (x – 2)2 + (y – 3)2 = 102 with (x – h)2 + (y – k)2 = r2, we can identify that h = 2, k = 3, and r = 10. So, we have got the coordinates of the center of circle which are (h, k) = (2, 3).
How to Find the Center of Circle with Two Points?
If the endpoints of the diameter of the circle are given, then to find the coordinates of the center we use the mid-point formula, because the center is the mid-point of the diameter of the circle. The steps to find the center of a circle with two points are given below:
- Step 1: Assume that the coordinates of the center of the circle are (h, k).
- Step 2: Use the midpoint formula which says that if (h, k) are the coordinates of the midpoint of a segment with endpoints (x1, y1) and (x2, y2), then (h, k) = [(x1 + x2]/2, [y1 + y2]/2).
- Step 3: Simplify it and get the coordinates of the center of the circle.
Let us take an example of a circle in which the endpoints of a diameter are given as (-2, 4), and (6, 16). Then, the coordinates of its center are:
(h, k) = [(-2 + 6)/2, (4 + 16)/2]
(h, k) = (4/2, 20/2)
(h, k) = (2, 10)
Therefore, the coordinates of the center of a circle with the endpoints of diameter are (2, 10).
☛ Related Articles
Check these interesting articles related to the concept of center of circle in geometry.
- Circle Formulas
- Sector of a Circle
- Circumference of Circle
- Area of Circle
FAQs on Center of Circle
What is the Center of Circle?
The center of a circle is the point where we place the tip of our compass while drawing a circle. It is the mid-point of the diameter of the circle. In a circle, the distance between the center to any point on the circumference is always the same which is called the radius of the circle.
What are the Coordinates for the Center of the Circle and the Length of the Radius?
The coordinates of the center of the circle represent the distance of the center point from the x-axis and y-axis respectively. It is generally denoted in the form of (h, k), where h and k represent the x and y coordinates respectively. The length of the radius is denoted by r. The coordinates of the center and the radius are related to each other in the form of an equation: (x – h)2 + (y – k)2 = r2.
What is the Center of a Circle Represented by the Equation (x – 5)
2 + (y + 6)2 = 42?
If we compare the given equation with the general equation of center of circle: (x – h)2 + (y – k)2 = r2, we can see that h = 5, k = -6, and r = √42. So, the center of the circle is at (5, -6).
How to Find Center of Circle?
To find the center of a circle, we can draw two parallel chords having the same length inside the circle.
Then, join the opposite ends of the chords. That point of intersection will be the center of the circle. The circle is also part of a conic section and the foci of the conic is the center of the circle.
How to Find Center of Circle with Endpoints of Diameter?
The center of a circle is the midpoint of the diameter. So, by using the midpoint formula, if the endpoints of the diameter are (a, b) and (c, d), then the coordinates of the center of circle are [(a + c)/2, (b + d)/2].
How to Find Radius and Center of Circle from Equation?
If the equation of a circle is given, then we can find its radius and center by comparing it with the general form of the equation: (x – h)2 + (y – k)2 = r2. We will find the values of h, k, and r. Then, (h, k) will be the coordinates of the center of circle and r will be the radius.
Basketball court markings: standards and norms
Author of the article
Khvatkov Dmitry
Consultant in the production of rubber coatings
Basketball field marking requirements are approved by the FIBA standard.
The site must be flat with a hard surface, free of bends, cracks and other obstacles. The accepted dimensions of the field are 28 m long and 16 m wide. By NBA standards, the field is slightly larger: 28.7 m (94′ ft) long and 15.3 m (50′ ft) wide.
Areas not intended for international competitions may differ from accepted standards (for public use, in schools or universities, etc.) and usually vary from 20 to 28 m in length and from 12 to 16 m in width.
Basketball Court Marking Standards
Basketball court markings are conventionally divided into 5 components:
- Boundary lines. They are located along the perimeter of the site and set its size. The lines that run along the field are called side lines, and those that are behind the baskets are called front lines.
- Central line. Divides the court in half parallel to the front lines.
- Central zone. It is a circle and is placed in the middle of the center line, and, accordingly, in the center of the entire field.
- Three-point line. It is a semi-ellipse and is located around the shields on both sides of the field. It limits the close range.
- Free throw line. It is located in front of the boards parallel to the front line and is limited on the sides by paint lines.
The standard line width is 5 cm. All outlines and lines must be of the same color (usually white) and be clearly visible from anywhere on the court.
Common lines
Common lines are used to limit the playing area of the court. The side lines (along the field) according to FIBA standards should be 28 m long, and the front lines – 16 m. For public areas, deviations from the accepted standards are allowed. Typically, basketball courts in schools or gyms are made from 20 m long and 12 m wide.
Central lines
The center line is parallel to the front and divides the field exactly in half. According to the standards – it should extend beyond the side lines by 15 cm on both sides.
In the middle of the center line there is a circle with a diameter of 3.6 m, which limits the central zone of the field. In this zone, the ball is played at the beginning of the game.
Three-Point Line
Three-Point Lines are located around the backboards on both sides of the field and consist of two straight lines 2.9 long9 m and a semicircle. Straight lines run perpendicular to the front at a distance of 0.9 m from the side lines. Despite the fact that visually the distance from the ring to the side of the three-point line seems to be less than to its central part, the distance from the backboard to any point is 6.75 m.
Penalty lines
Penalty lines limit the nearest area at the backboard. They consist of a trapezoid and a free throw zone.
Despite the name, the “trapezium” is a rectangle (until 2009year it really was a trapezoid), which is located under the shield. Its dimensions are 5.8 meters long and 4.9 meters wide. The shield is located at a distance of 1.
575 m from the end line in the middle of the site. In front of the backboard, at a distance of 1.25 m, there is a semicircle that limits the area for picking up the ball.
At a distance of 4.225 meters from the backboard, the trapeze zone ends and the free throw zone begins. It is a semicircle with a diameter of 3.6 m (like the central circle).
Paint zone lines
These lines are serifs on both sides of the trapezoid (parallel to the sidelines). They limit the areas for players who are fighting for the ball during a free throw.
Zones on the basketball field
The basketball court is divided into zones using markings. Each zone has its own specific rules.
Center circle
The center circle is used as a separate kick-off area at the start of the game. One representative from each team stand in a circle from their side and fight for the ball in a jump, after it is dropped by the referee. All players are exclusively on their side of the field, except for one who rebounds on the opponent’s side.
Neutral zone
The peculiarity of this zone is that as soon as the player of the attacking team with the ball crosses the center line and is on the side of the opponent, he cannot pass the ball to the player of his team who is on the other side of the field (i.e. behind center line on your side).
Three-point zone
The three-point line limits the near zone of the shot. Hitting the basket from outside the basket brings the team three points. If the throw was made inside the zone, then it brings two points.
Three-second zone
This is the zone in close proximity to the ring. It is called three-second, since the player of the attacking team cannot be in it for more than three seconds. Most balls are thrown in this zone, so when attacking, it provides maximum protection.
Free throw area
In controversial situations, a free throw is provided from this area. The player of the attacking team must score the ball without stepping over the line of the trapezoid.
At the same time, the players of both teams are not in the three-second zone. They take up positions along the paint lines on the sides of the trapezoid and may not step outside the lines until the free throw shooter has shot the ball.
How to mark a basketball field?
Basketball field markings, whether it is an international competition court or an open-air amateur field, are best applied using special equipment. This will ensure the long life of the coating, the lines will not clog and will promote fair play.
You can order the marking of a basketball court in Moscow and the Moscow region from Rezkom. We will measure the premises and develop a design project for the field so that it complies with generally accepted rules and is convenient for operation. For more details, you can contact our manager by phone 8-495-64-24-111.
Ivanov V. Central circle
- pdf format
- size 8.9 MB
- added
December 26, 2010
M .
: “Physical culture and sport”, 1973. – 256 p.
Famous football player, forward of the Moscow “Torpedo” and the national team
team of the USSR Valentin Ivanov performed in those happy for our
football years when the team won the titles of the champion of the Olympic
games and the European Cup. Partners and comrades of Ivanov on the national team and
“Torpedo” were Yashin and Netto, Simonyan and Bobrov, Shesternev and
Metreveli, Meskhi and Monday, Muntyan and Byshovets. Ivanov is now
head of the Torpedo team. In the book, he talks about his journey to
football, about your friends, partners, rivals, and all this
material serves him to talk about the most pressing problems
modern Soviet football, which he tries to look at
simultaneously from the positions of the player and the coach.
See also
Directory
- djvu format
- size 5.52 MB
- added
August 31, 2011
M.
: “Moskovskaya Pravda”, 1984. – 64 p. The book is dedicated to the championship of the Soviet Union in 1984 (second round), as well as the European Championship 1984 years in France.
Directory
- djvu format
- size 1.29 MB
- added
September 08, 2011
M.: “Moskovskaya Pravda”, 1959. – 97 p. The reference book is devoted to the past football championship of the Soviet Union in 1958, as well as the upcoming USSR championship 1959 years old (first round).
- doc format
- size 2.37 MB
- added
December 25, 2010
M.: “Military publishing house of the Ministry of Defense of the USSR”, 1973.
– 147 p. CSKA… Central Army Sports Club. How widespread is the fame, how immense is the popularity of this club! He is well known to all athletes of our country. His sports life is closely followed by a huge mass of sports fans. CSKA has gained respect in sports circles around the world. For most fans of army sports, the concept of CSKA is associated primarily with hockey and f…
- pdf format
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October 03, 2011
Leningrad: “Lenizdat”, 1987. – 112 p., 2nd ed., revised. and additional Honored Master of Sports L.G. Ivanov, a popular goalkeeper in the past, who defended the gates of the Leningrad Zenit for eighteen years, talks about himself, about his friends – the players of Zenit, the national teams of Leningrad and the Soviet Union, about the most memorable matches, about how the best traditions are preserved and multiplied Leningrad football. The book is intended for a wide range of readers…
Directory
- pdf format
- size 13.22 MB
- added
November 21, 2011
Minsk: Publishing House of the Central Committee of the CPB, 1954. – 59 p. The directory-calendar shows the results of the national football championship of 1954 in class “A” for 1 round, and also shows the calendar of games for the second round of the USSR championship in classes “A” and “B” of season 1954 years old.
Directory
- pdf format
- size 9.42 MB
- added
August 31, 2011
M .: “Central Stadium named after V.
I. Lenin”, 1991. – 62 p. Results of the penultimate 53rd championship of the USSR in 1990 (figures, statistics, results, teams, etc.). The story of the USSR Cup 1989-1990 A smooth transition to coverage of the future national championship in 1991 (as it turned out – the last one), as well as the USSR Cup (lineups, calendar of games, etc.). A lot of information about the USSR national team, a review of its performances on the international arena in 1990…
Directory
- pdf format
- size 8.92 MB
- added
29December 2011
M.: “Moskovskaya Pravda”, 1975. – 96 p. The reference book is dedicated to the 1975 Soviet Union football championship (first round), the 1975 USSR Cup, as well as international competitions with the participation of Soviet teams.
Directory
- pdf format
- size 12. 87 MB
- added
September 04, 2011
87 MBM.: “Moskovskaya Pravda”, 1981. – 96 p. The brochure is dedicated to the championship of the Soviet Union in 1981 (second round), as well as the USSR Cup and other football events in the country and the world.
Directory
- pdf format
- size 12.58 MB
- added
September 04, 2011
M.: “Moskovskaya Pravda”, 1983. – 95 p. The brochure is dedicated to the championship of the Soviet Union in 1983 (second round), as well as the USSR Cup, the Spartakiad and other football events in the country and the world.
- pdf format
- size 11.36 MB
- added
December 27, 2010
M.
