The formula for the circumference of a circle is c = 2π.

r

The area formula of a circle is s = π.

r？

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A triangle has a unique circumscribed 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;

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.

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The intersection of two tangent circles (intersection: a straight line with two centers connected)

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The midpoint m of the chord PQ on the circle O. If the intersection point m is the intersection of two chords AB, CD, AD and BC with PQ on X and Y respectively, then M is the midpoint of XY.

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If two circles intersect, the line segment (or straight line) connecting the centers of the two circles bisects the common chord vertically.

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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.

The degree of the chord tangent angle is equal to half the degree of the arc it encloses.

The degree of the angle inside a circle is equal to half of the sum of the degrees of the arcs subtended by this angle.

The degree of the outer angle of a circle is equal to half the difference between the degrees of two arcs cut by this angle.

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Tangent secant theorem

The tangent and secant of a circle intersect at point p, tangent intersects at point c, and secant intersects at point a.

2 o'clock

Then there is PC 2 = pa Pb.

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secant theorem

Similar to the cutting line theorem

The two secant intersect at point p, and secant m intersects at A 1.

At two o'clock direction B 1, the secant n intersects A2.

B2 at two o'clock.

Then pa 1 pb 1 = pa2 pb2.

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The standard equation of a circle: In the plane rectangular coordinate system, the standard equation of a circle with a radius of R and a center of point O(a, b) is (x-a) 2+(y-b) 2 = r 2.

General equation of a circle: after expanding the standard equation of a circle, shifting terms and merging similar terms, the general equation of a circle can be obtained as x 2+y 2+dx+ey+f = 0 (where d 2+e 2-4f > 0). Compared with the standard equation, in fact, D=-2a, E=-2b and f = a 2+b 2-r 2. The center coordinates of the circle are (-D/2, -E/2) and the radius r = 0.5 √ d 2+e 2-4f.

Parametric equation of circle: The parametric equation of a circle with point O(a, b) as the center and R as the radius is

x=a+r*cosθ，

y=b+r*sinθ，

(where θ is a parameter)

Endpoint formula of a circle: If two points A (A 1, B 1) and B (A2, B2) are known, the equation of a circle with line segment AB as its diameter is

(x-a 1)(x-a2)+(y-b 1)(y-B2)= 0

The eccentricity of a circle is e=0, and the radius of curvature of any point on the circle is r.

Go through a circle

The tangent equation of point M(a0, b0) on x 2+y 2 = r 2 is

a0*x+b0*y=r^2

A point M(a0, b0) outside the circle (X 2+Y 2 = R 2) leads to two tangents of the circle, which are A and B, so is the equation of the straight line where A and B are located.

a0*x+b0*y=r^2

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Symmetry of circle: A 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.

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Properties and theorems of central angle and central angle.

In the same circle or in the same circle, if the distance between two central angles, two peripheral angles, two sets of arcs, two chords and one of the two chords is equal, the corresponding other groups are equal.

An arc subtends a circumferential angle equal to half the central angle it subtends.

The circumferential angle of the diameter is a right angle. The chord subtended by a 90-degree circle angle is the diameter.

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.