Theorem: The diameter perpendicular to the chord bisects the chord and bisects the two arcs opposite the chord.
Inference 1:
(1) bisects the diameter of the chord (not the diameter) perpendicular to the chord and bisects the two arcs opposite to the chord;
(2) The perpendicular line of the chord passes through the center of the circle and bisects the two arcs opposite to the chord;
(3) bisect the diameter of an arc opposite to the chord, bisect the chord vertically, and bisect another arc opposite to the chord.
Inference 2: The arcs between two parallel chords of a circle are equal.
Note: (1) Vertical Diameter Theorem and its deduction are important basis for proving the equality of line segments, arcs and angles. When solving the problem of chord in a circle, the diameter perpendicular to the chord is often used as an auxiliary line.
(2) The vertical diameter theorem can be rewritten as: If a straight line is perpendicular to a chord and passes through the center of the circle, the straight line bisects the chord and bisects the two arcs opposite to the chord. There are four conditions: the straight line is perpendicular to the chord, the straight line bisects the chord, the straight line crosses the center of the circle, and the straight line bisects the arc opposite to the chord. Its three inferences can be regarded as "if two of the four conditions are true, the other two are also true". This is how the vertical diameter is understood and remembered.
Definition: If the diameter of a circle is perpendicular to the chord, then the diameter bisects the chord and bisects the arc opposite to the chord.
Inference 1: bisect the chord (not the diameter). The diameter of the chord is perpendicular to this chord and bisect the two arcs opposite to this chord.
Inference 2: The perpendicular line of a chord passes through the center of the circle and bisects the arc opposite to this chord.
Inference 3: The diameter of the arc bisecting a chord bisects this chord vertically and bisects another arc opposite this chord. Inference 4: In the same or equal circle, the arcs sandwiched by two parallel chords are equal.
(The theoretical basis of proof is the above five theorems) Edit this paragraph of proof.
As shown in the figure, in ⊙O, DC is the diameter, AB is the chord, AB⊥DC,AB and CD intersect at E, proving AE=BE, arc AC= arc BC, arc AD= arc BD. The vertical diameter theorem proves that the graph connects OA and OB.
OA and OB are radii.
∴OA=OB
△ OAB is an isosceles triangle.
∵AB⊥DC
∴AE=BE,∠AOE=∠BOE (isosceles triangle connected by three lines)
∴ arc AD= arc BD, ∠AOC=∠BOC.
∴ Arc AC= Arc BC Edit this explanation.
The vertical diameter theorem is also called "5-2-3" theorem.
Meaning: ①CD⊙O, diameter AB is chord; ②cd⊥ab； ③AE = BE； ④ arc AD= arc BD; ⑤ Arc AC= Arc BC If any two of the above five conditions are satisfied, the other three conditions are also established.
The following is the inference of this paragraph.
Inference 1: The diameter (not the diameter) of the bisecting chord is perpendicular to this chord and bisects the two arcs opposite to this chord.
Inference 2: The perpendicular line of a chord passes through the center of the circle and bisects the arc opposite to this chord.
Inference 3: The diameter of the arc bisecting a chord bisects this chord vertically and bisects another arc opposite this chord. Inference 4: In the same or equal circle, the arcs sandwiched by two parallel chords are equal.
(The theoretical basis of proof is the above five theorems)
However, in the problem that there is no need to write a proof process, you can use the following methods to judge:
As long as a straight line meets any two of the following five conditions, the other three conclusions can be deduced.
The best arc to bisect the chord.
The lower arc bisecting the chord (the first two together are: the two arcs bisecting the chord)
Bisect a chord (not a diameter)
Perpendicular to the chord
Through the center of the circle,
The diameter perpendicular to the chord bisects the chord and the two arcs it faces.