What are the main theorems about circles in geometry?

Main Circle Theorems in Geometry

Here are the main theorems about circles you should know:

• Angle at the Center Theorem

The angle subtended by an arc at the center is twice the angle subtended at the circumference:

• If arc AB subtends angle θ at the center O, then arc AB subtends θ/2 at any other point on the circle.
• Angles in the Same Segment Are Equal

Angles in the same segment, subtended by the same chord, are equal.

• Cyclic Quadrilateral Theorem

The opposite angles of a cyclic quadrilateral (a quadrilateral inscribed in a circle) sum to 180°.

• Tangent-Secant Theorem

A tangent and a radius meet at a right angle (90°). Also, tangents from a common external point are equal in length.

• Alternate Segment Theorem

The angle between a tangent and a chord through the point of contact equals the angle in the alternate segment.

Example:
If AB is a chord and P is a point on the circumference on the same side as the center, then angle APB = ½ × central angle AOB.

Takeaways:

• Knowing the main circle theorems helps in solving diverse geometry problems.
• These rules connect angles, chords, tangents, and segments within and outside circles.