What is a circle as a geometric figure: basic properties and characteristics

In general, imagine what a circle is, look at the ring or hoop. You can also take a round glass and a cup, put it upside down on a sheet of paper and circle it with a pencil. With a multiple increase, the resulting line will become thick and not quite even, and its edges will be blurred. A circle as a geometric figure does not have such a characteristic as thickness.

Circumference: definition and basic means of description

A circle is a closed curve consisting of a set of points located in the same plane and equidistant from the center of the circle. The center is in the same plane. As a rule, it is denoted by the letter O.

The distance from any of the points of the circle to the center is called the radius and is denoted by the letter R.

If you connect two any points of a circle, then the resulting segment will be called a chord. The chord that passes through the center of the circle is the diameter denoted by the letter D. The diameter divides the circumference into two equal arcs and is twice the length of the radius along the length. Thus, D = 2R, or R = D / 2.

Properties of chords

1. If through two arbitrary points of a circle hold a chord, and then perpendicular to the last one - a radius or a diameter, then this segment will break both the chord and the arc cut off by it into two equal parts. The converse is also true: if the radius (diameter) divides the chord in half, then it is perpendicular to it.
2. If two parallel chords are drawn within the same circle, then the arcs cut by them, as well as those enclosed between them, will be equal.
3. We draw two chords PR and QS that intersect within the circle at the point T. The product of the segments of one chord will always be equal to the product of the segments of the other chord, that is, PT x TR = QT x TS.

Circumference: general concept and basic formulas

One of the basic characteristics of this geometric figure is the circumference. The formula is derived using such quantities as radius, diameter and constant "π", reflecting the constancy of the ratio of the circumference to its diameter.

Thus, L = πD, or L = 2πR, where L is the circumference, D is the diameter, and R is the radius.

The formula for the length of a circle can be considered as the initial one for finding the radius or diameter for a given circumference length: D = L / π, R = L / 2π.

What is a circle: the basic postulates

1. The line and the circle can be located on the plane as follows:

• Not have common points;
• Have one common point, while the straight line is called a tangent: if you draw a radius through the center and the point of tangency, it will be perpendicular to the tangent;
• Have two common points, while the straight line is called a secant point.

2. Through three arbitrary points lying in one plane, one can draw no more than one circle.

3. Two circles can touch only one point, which is located on the segment connecting the centers of these circles.

4. For any rotations relative to the center, the circle passes into itself.

5. What is a circle in terms of symmetry?

• The same curvature of the line at any of the points;
• Central symmetry about the point O;
• Mirror symmetry with respect to diameter.

6. If we construct two arbitrary inscribed angles that are supported by the same arc of a circle, they will be equal. The angle, resting on an arc equal to half the circumference, that is, cut off by a chord-diameter, is always 90 °.

7. If we compare closed curves of the same length, it turns out that the circle delimits a section of the plane of the largest area.

A circle inscribed in a triangle and described near it

The idea of what a circle is will be incomplete without describing the features of the relationship between this geometric figure and triangles.

1. When constructing a circle inscribed in a triangle, its center will always coincide with the point of intersection of the bisectors of the angles of the triangle.
2. The center of the circle described near the triangle is located at the intersection of the median perpendiculars to each side of the triangle.
3. If we describe a circle around a right triangle, then its center will be in the middle of the hypotenuse, that is, the latter will be the diameter.
4. The centers of the inscribed and circumscribed circles will be at one point if the basis for the construction is an equilateral triangle.

Basic statements about the circle and quadrangles

1. Around a convex quadrilateral, one can describe a circle only when the sum of its opposite internal angles is 180 °.
2. A circle inscribed in a convex quadrilateral can be constructed if the sum of the lengths of its opposite sides is the same.
3. You can describe a circle around a parallelogram if its angles are straight.
4. You can enter a circle in a parallelogram if all its sides are equal, that is, it is a rhombus.
5. Construct a circle through the corners of the trapezoid only if it is isosceles. In this case, the center of the circumscribed circle will be located at the intersection of the symmetry axis of the quadrilateral and the median perpendicular drawn to the side.