1. Definition of a circle: A circle is a set of all points with the same distance from a plane to a fixed point. This fixed point is called the center of the circle and the distance is called the radius.
2. Tangent Theorem: If a straight line is tangent to a circle, then the straight line is perpendicular to the radius passing through the tangent point.
3. Chord length theorem: In a circle, the product of two intersecting chords is equal to the product of the arc between them. That is, AB×CD=AD×BC, where a, b, c and d are the ends of the chord.
4. Vertical diameter theorem: In a circle, the chord perpendicular to the diameter bisects the arc opposite to the diameter.
5. Theorem of the sum of triangle internal angles: In a circle, the sum of triangle internal angles with the center of the circle as the vertex is equal to 180 degrees.
6. Quadrilateral Interior Angle Sum Theorem: In a circle, the quadrilateral interior angle sum with the center of the circle as the vertex is equal to 360 degrees.
7. The circle angles of the same arc are equal: in the same circle or equal circle, the circle angles of the same arc are equal.
8. The central angle of the same arc is equal to twice the central angle: in the same circle or equal circle, the central angle of the same arc is equal to twice the central angle of the same arc.
9. Properties of concentric circles: If two circles have the same center, the difference of their radii is equal to the distance between their centers.
10. Sector area formula: The area of a sector can be calculated by the following formula: A=(θ/360)×r_, where a is the sector area, θ is the central angle (in radians) of the sector relative to it, and r is the radius.
The above are just some basic theorems about circles, and there are many other more complicated theorems and properties that can be further explored and applied.