Cyclic quadrilaterals

Can you use the key properties of cyclic quadrilaterals to find quick solution paths on certain AMC problems?

Definitions

cyclic quadrilateral: A four-sided figure whose vertices are all on the circumference of some circle.
circumscribed circle (or circumcircle): The circle whose circumference contains all four vertices of the cyclic quadrilateral.
concyclic: The adjective describing the vertices of a cyclic quadrilateral; this is the property that makes a quadrilateral cyclic.

The interior angles of cyclic quadrilaterals have certain important properties. The opposite ones sum to $18 0^{o}$ , for example. But there are important lesser-known properties as well, involving the angles formed by the diagonals of the cyclic quadrilateral.

The lengths of the diagonals are also constrained in a super-interesting way: their product is equal to the sum of the products of the opposite sides of the quadrilateral. (This seems a little crazy until you consider how this might be related to the Pythagorean theorem.)

Even the area of a c.q. has a formula associated with it. (It’s like Heron’s formula for triangles: it gives the area of a c.q. in terms of the lengths of its sides only.)

These facts are all covered in the quizzes for this section.