1. **State the problem:** We will list and explain all 9 circle theorems, which are fundamental properties relating angles, chords, tangents, and segments in a circle. 2. **Theorems:** 1. **Angle at the Center Theorem:** The angle subtended at the center of a circle by an arc is twice the angle subtended at the circumference by the same arc. 2. **Angle in a Semicircle Theorem:** The angle subtended by a diameter at the circumference is a right angle ($90^\circ$). 3. **Angles in the Same Segment Theorem:** Angles subtended by the same chord and on the same side of the chord are equal. 4. **Opposite Angles of a Cyclic Quadrilateral Theorem:** The sum of the opposite angles of a cyclic quadrilateral is $180^\circ$. 5. **Alternate Segment Theorem:** The angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment. 6. **Tangent to a Circle Theorem:** A tangent to a circle is perpendicular to the radius drawn to the point of contact. 7. **Equal Tangents Theorem:** Tangents drawn from an external point to a circle are equal in length. 8. **Chord Bisector Theorem:** The perpendicular from the center of a circle to a chord bisects the chord. 9. **Radius to a Chord Theorem:** The line from the center of the circle to the midpoint of a chord is perpendicular to the chord. 3. **Explanation:** Each theorem describes a specific relationship involving angles, lengths, or perpendicularity in circles. These theorems are used to solve many geometry problems involving circles. 4. **Summary:** Knowing these 9 theorems allows you to analyze and solve problems involving circle geometry effectively. No algebraic formula is needed here, but understanding these properties is essential for circle geometry.