Related concepts of 1. circle
Circle, center, radius, chord, diameter, arc, semicircle, upper arc, lower arc, chord center distance, equal arc, equal circle, concentric circle, bow and bow height.
Description:
(1) The diameter is a chord, but the chord is not necessarily a diameter, and the diameter is the longest chord in a circle.
(2) A semicircle is an arc, but an arc is not necessarily a semicircle.
(3) Equal arcs can only be arcs in the same circle or in the same circle. Without the condition of "same circle or equal circle", there is no equal arc.
(4) Equal-length arcs must be equal in length, but they are not necessarily equal.
2. The positional relationship between a point and a circle
Note: the positional relationship between a point and a circle corresponds to the quantitative relationship between the distance from the point to the center of the circle and the radius, that is, knowing the positional relationship can determine the quantitative relationship; Knowing the quantitative relationship, we can also determine the positional relationship.
3. Angle related to the circle
Central angle and external angle
Note: These two angles related to the circle can be compared from the position of the vertex of the (1) angle; (2) The positional relationship between an angle and both sides of a circle can be grasped from two aspects.
Supplement: if the vertex of an angle is in a circle, it is called the inner angle of the circle, and the central angle is a special inner angle of the circle; If the vertex of an angle is outside the circle and both sides of the angle intersect the same circle, such an angle is called an external angle.
4. Related properties of circles
Determination of (1) circle
& lt 1 & gt； The center of the circle determines the position of the circle, and the radius determines the size of the circle.
& lt2> Three points that are not on the same straight line determine a circle.
(2) Symmetry of the circle
& lt 1 & gt； A circle is an axisymmetric figure, and any straight line passing through the center of the circle is its symmetry axis.
& lt2> circle is a figure with a symmetrical center, and the center of the circle is its symmetrical center.
Note: There are countless symmetry axes of a circle and only one center of symmetry. If the circle rotates at any angle around the center of the circle, it can coincide with the original figure, that is, the circle has rotation invariance.
(3) vertical diameter theorem
If a straight line has (1) a lower arc (5) bisecting the chord (4) perpendicular to the chord (3) through the center (2) and an upper arc bisecting the chord, the straight line has three other properties, namely:
Vertical diameter theorem: (1)(2) (3)(4)(5)
Inference 1: (1) (3) (2) (4) (5)
(2)(3) ( 1)(4)(5)
(1)(4) (or (5)) (2)(3)(5) (or (4))
(1)(3) (2)(4)(5) means "bisect the diameter (not the diameter) of the chord perpendicular to the chord and bisect the two arcs opposite to the chord", where the chord must be a non-diameter chord. If the chord is a diameter, the two diameters are not necessarily perpendicular to each other.
Inference 2: The arcs sandwiched by two parallel chords of a circle are equal.
Note: When solving problems related to circles, the following auxiliary lines are often cited:
(1) Radius of the endpoint and center of the chord.
(2) make the chord center distance
(3) the midpoint of the chord connecting the center of the circle (when encountering the midpoint of the chord)
(4) Connect the center of the circle with the midpoint of the arc (when encountering the midpoint of the arc)