Euclidean geometry

Theorem 1: Line from Center Perpendicular to Chord

A line drawn from the center of a circle perpendicular to a chord bisects the chord; conversely, the line drawn from the center of a circle to the midpoint of a chord is perpendicular to the chord.

Theorem 2: Angle at Center

The angle subtended by an arc at the center of a circle is double the angle subtended by the same arc at the circumference of the circle.

Theorem 3: Angles in Same Segment

Angles in the same segment of a circle are equal.

Theorem 4: Cyclic Quadrilateral

The opposite angles of a cyclic quadrilateral are supplementary (add up to 180^{\circ}).