Cyclic quadrilaterals (cloned) | Geometry (cloned)

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.

What to do

If everything above was old news to you, then this section should be no problem. Click “complete” below, and we’ll add all the relevant quizzes to your deck. Then, during your short daily practice, we’ll occasionally quiz you on this knowledge in a way that etch it into your memory for good.
If some or all of this was new to you, then you should still click “complete” below, and add it to your quizzes, but I recommend against adding anything else for today. Instead, work with these new cards for today, and consider adding more fresh knowledge tomorrow. You might also do a bit of research to refresh (or learn for the first time) the rules given above.