Circle Theorems

1. Angle at the Centre  

   - The angle subtended at the centre of a circle is twice the angle subtended at the circumference by the same arc.  

2. Angles by the Same Arc  

   - Angles subtended at the circumference by the same arc are always equal.  

3. Angle in a Semi‑circle 

   - The angle formed in a semi‑circle is always a right angle (90°).  

4. Cyclic Quadrilateral  

   - In a cyclic quadrilateral (all vertices lie on a circle), opposite angles add up to 180°.  

5. Tangent and Radius  

   - The angle between a tangent and the radius drawn to the point of contact is 90°.  

6. Tangents from a Point 

   - From any point outside a circle, exactly two tangents can be drawn.  

   - These tangents are of equal length.  

7. Alternate Segment Theorem  

   - The angle between a tangent and its chord is equal to the angle in the alternate segment.  

8. Equal Chords  

   - Equal chords of a circle are equidistant from the centre.  

9. Perpendicular Bisector of a Chord 

   - The perpendicular bisector of a chord always passes through the centre of the circle.