Condition for Four Points to Lie on a Circle

If two points and are on the same side of line , and if

then the four points and lie on the same circle.

If four points and lie on the same circle, then , which corresponds to the inscribed angles subtended by arc .
However, if points and are on opposite sides of line , even if , we cannot conclude that the four points lie on the same circle.

Properties of a Cyclic Quadrilateral

1. In a cyclic quadrilateral (a quadrilateral inscribed in a circle), the sum of the measures of opposite angles is .
   
2. In a cyclic quadrilateral, the measure of an exterior angle is equal to the measure of the opposite interior angle.
   

Conditions for a Quadrilateral to be Inscribed in a Circle

1. A quadrilateral whose opposite angles sum to can be inscribed in a circle.
2. A quadrilateral in which the measure of an exterior angle equal to the opposite interior angle can be inscribed in a circle.

Squares, rectangles, and isosceles trapezoids always satisfy the condition that the sum of the opposite angles is , so they can always be inscribed in a circle.