When drawing a circle with a compass, the point where the needle tip is located is called the center of the circle, which is generally represented by the letter O. The line segment connecting the center of the circle with any point on the circle is called the radius, which is generally represented by the letter R. The length of the radius is the distance between the two corners of the compass. The line segment passing through the center of the circle and with both ends on the circle is called the diameter, which is generally represented by the letter D.
A circle is a curved figure on a plane and an axisymmetric figure. Its axis of symmetry is the straight line where the diameter lies, and the circle has countless axes of symmetry.
History of the circle
A circle is a seemingly simple shape, but in fact it is very wonderful. The ancients first got the concept of circle from the sun and the moon on the fifteenth day of the lunar calendar. /kloc-Neanderthals 0/8000 years ago used to drill holes in animal teeth, gravel and stone beads, some of which were round. In the pottery age, many pottery were round. Round pottery is made by putting clay on a turntable. When people start spinning, they make round stone spindles or ceramic spindles. The ancients also found it easier to roll when carrying logs. Later, when they were carrying heavy objects, they put some logs under big trees and stones and rolled them around, which was of course much more labor-saving than carrying them.
About 6000 years ago, Mesopotamia made the world's first wheel-a round board. About 4000 years ago, people fixed round boards under wooden frames, which was the original car.
You can make a circle, but you don't necessarily know its nature. The ancient Egyptians believed that the circle was a sacred figure given by God. It was not until more than two thousand years ago that China's Mozi (about 468- 376 BC) gave the definition of a circle: a circle, a circle of equal length. It means that a circle has a center and the length from the center to the circumference is equal. This definition is 100 years earlier than that of the Greek mathematician Euclid (about 330 BC-275 BC).
The concept of circle
1. A point set whose distance to a fixed point is equal to a fixed length is called a circle. This fixed point is called the center of the circle and is usually represented by the letter "O".
2. The straight line connecting the center of the circle and any point on the circumference is called radius, which is usually represented by the letter "R".
3. The line segment with two ends on the circumference passing through the center of the circle is called the diameter, which is usually represented by the letter "D".
A line segment connecting any two points on a circle is called a chord. In the same or equal circle, the longest chord is the diameter.
5. The part between any two points on a circle is called an arc. An arc larger than a semicircle is called an optimal arc and is represented by three letters. An arc smaller than a semicircle is called the lower arc and is represented by two letters. A semicircle is neither an upper arc nor a lower arc.
The letter stands for circle-⊙; Radius —r or r (the letter represented by the radius of the outer ring in the ring); Arc; Diameter d or d;
Sector arc length-l; Circumference-c; Area -S
Nature of circle
The (1) circle is an axisymmetric figure, and its symmetry axis is any straight line passing through the center of the circle. A circle is also a central symmetric figure, and its symmetric center is the center of the circle. ?
Vertical diameter theorem: the diameter perpendicular to the chord bisects the chord and bisects the two arcs opposite the chord. Inverse theorem: bisecting the diameter of a chord (not the diameter) is perpendicular to the chord and bisecting the two arcs opposite the chord.
⑵ The properties and theorems of central angle and central angle.
(1) In the same circle or the same circle, if one of two central angles, two peripheral angles, two sets of arcs, two chords and the distance between two chords is equal, their corresponding other groups are equal respectively.
(2) The arc faces a circumferential angle equal to half of its central angle.
The circumferential angle of the diameter is a right angle. The chord subtended by a 90-degree circle angle is the diameter.
Calculation formula of central angle: θ = (l/2π r) × 360 =180l/π r = l/r (radian) (angle system and arc system: 360 = 2π).
That is, the degree of the central angle is equal to the degree of the arc it faces; The angle of a circle is equal to half the angle of the arc it faces.
(3) If the length of an arc is twice that of another arc, then the angle of circumference and center it subtends is also twice that of the other arc.
⑶ Properties and theorems about circumscribed circle and inscribed circle.
① A triangle has a unique circumscribed circle (∵ three points define a circle).
Circle and inscribed circle. The center of the circumscribed circle is the intersection of the perpendicular lines of each side of the triangle, and the distances to the three vertices of the triangle are equal;
(2) The center of the inscribed circle is the intersection of the bisectors of the inner angles of the triangle, and the distances to the three sides of the triangle are equal.
③R=2S△÷L(R: radius of inscribed circle, S△: area of triangle, l: perimeter of triangle).
(4) the intersection of intersecting lines of two tangent circles (intersecting line: a straight line with two centers connected)
⑤ The midpoint m of PQ on the upper chord of circle O, and the intersection point m is defined as the intersection of two chords AB, CD, AD and BC with PQ on X and Y respectively, then M is the midpoint of XY.
(4) If two circles intersect, the line segment (or straight line) connecting the centers of the two circles vertically bisects the common chord.
(5) The degree of the chord tangent angle is equal to half the degree of the arc it encloses.
(6) The degree of the angle inside a circle is equal to half of the sum of the degrees of the arcs subtended by the angle.
(7) The degree of the outer angle of a circle is equal to half of the difference between the degrees of two arcs cut by this angle.
(8) The perimeters are equal, and the area of a circle is larger than that of a rectangle, a square or a triangle.