The vertices of the three angles are on the circle, and the parts on both sides of them inside the circle are the chords of the circle.

Circle Angle Theorem

The measure of a circumferential angle is equal to half the measure of the arc it subtends

The circumferential angles subtended by congruent arcs or congruent arcs are equal

The circumferential angle subtended by the diameter is a right angle, and the chord subtended by the 90° circumferential angle is the diameter.

Pay attention to the use of auxiliary lines

The circle passing through each vertex of a triangle is called the circumcircle of the triangle

The center of the circumcircle is the intersection point of the perpendicular bisectors of the three sides of the triangle, which is called the circumcenter of the triangle.

This triangle is called the inscribed triangle of the circle

Complementary diagonals of a quadrilateral inscribed in a circle

Any exterior angle of a quadrilateral inscribed in a circle is equal to its opposite interior angle

outer center

heart

center of gravity

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Pay attention

In congruent circles or congruent circles, equal central angles subtend arcs that are equal and subtend chords that are equal.

A semicircle is an arc, but an arc is not necessarily a semicircle

An arc of one degree? significance

In the same circle or equal circles, two arcs that can overlap are called equal arcs.

The radius is the chord, but the chord is not necessarily the radius

The line segment connecting any two points on a circle is called a chord

In congruent circles or equal circles, if one set of quantities in two central angles, two arcs, and two chords are equal, then the remaining sets of quantities corresponding to them are equal respectively.

The diameter perpendicular to the chord bisects the chord and bisects the two arcs subtended by the chord.

The diameter that bisects the chord (not the diameter) is perpendicular to the chord and bisects the two arcs subtended by the chord

When d<r, the point is inside the circle

When d=r, the point is on the circle

When d>r, the point is outside the circle

Two circles with equal radii are called equal circles. Two equal circles can overlap

The chord passing through the center of the circle is called the diameter

The fixed point is the center of the circle, and the fixed length is the radius. A circle with point O as the center is denoted as ⊙O and pronounced as "circle O"