1. fillet theorem: the fillet opposite to the same arc is equal to half of the fillet opposite to it.

By observing ∠B, ∠P and ∠AOC, we can get ∠B=∠P= 1/2∠AOC.

2. Theorem of Central Angle: In the same circle or an equal circle, an equal central angle has equal chord, equal arc and equal chord center distance.

By observing ∠B, ∠P, ∠AOC, arc AC, chord AC and chord center distance OS, the above conclusions can be drawn.

Remarks: Of the four conclusions of the central angle theorem, as long as we know that 1 is equal, we can deduce the other three conclusions.

3. Tangent judgment theorem: A straight line passing through the outer end of the radius and perpendicular to the radius is a tangent.

The above conclusions can be drawn by observing OC and CT.

4. Tangent property theorem: the tangent is perpendicular to the radius of the tangent point.

The above conclusions can be drawn by observing TC and co.

5. Tangent length theorem: Two tangents of a circle are drawn from a point outside the circle, and their tangents are equal in length.

The above conclusions can be drawn by observing TC and TA.

6. Intersecting chord theorem: When two chords intersect in a circle, the product of two line segments is equal.

Observing △PAE and △BCE, because ∠P=∠B, ∠PEA=∠BEC, ∠PAE=∠BCE, the corresponding edges are directly proportional to PE: Be = EA: EC.

7. Secant theorem: The tangent and secant of a circle are drawn from a point outside the circle, and the length of the tangent is the median of the ratio of the two line segments where the secant intersects the circle.

By observing the tangent FC and secant FA, as well as △FCD and △FAC, it is easy to prove that △FCD∽△FAC, the corresponding edge is proportional to FC: FA = FD: FC, and FC 2 = FA FD is obtained by cross multiplication.

8. Secant theorem: Two secants of a circle are drawn from a point outside the circle, and the product of the lengths of the two lines from that point to the intersection of each secant and the circle is equal.

Suppose that the other secant passing through point F is FA', then there is FC 2 = FA' FD' in the same way, and since FC 2 = FA FD, FA' FD' = FA FD, that is, the product of the lengths of two straight lines from point F to the intersection of each secant and the circle is equal.

9. Common Chord Theorem of Two Circles: The line connecting the centers of two circles is vertical and bisects the common chord of these two circles.

Observing △OAC, chord AC and chord center distance OS, OS can easily divide AC vertically. Suppose AC is also the chord of the circle O', and O' must bisect AC vertically, so OO' bisects AC vertically, that is, the center lines of the two circles are vertical and bisect the common chord of the two circles.

10. Chord angle theorem: The angle at which the vertex intersects the circle on one side and the other side is tangent to the circle is called the chord angle, and the chord angle is equal to the circumferential angle of the arc it encloses.