Cyclic quadrilaterals

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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.

Cyclic Quadrilaterals: Definitions

Four vertices lie on a single circle (concyclic)
Circumscribed circle (circumcircle) passes through all vertices

Angle Properties

Opposite interior angles sum to $18 0^{\circ}$
Angles formed by diagonals have special relationships

Diagonal Lengths

Product of diagonals equals sum of products of opposite sides
- $A C \cdot B D = A B \cdot C D + A D \cdot BC$

Area Formula

Area can be found using a formula similar to Heron’s, based on side lengths only

Cyclic quadrilaterals

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.

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.

Key points

Cyclic Quadrilaterals: Definitions

Four vertices lie on a single circle (concyclic)
Circumscribed circle (circumcircle) passes through all vertices

Angle Properties

Opposite interior angles sum to $18 0^{\circ}$
Angles formed by diagonals have special relationships

Diagonal Lengths

Product of diagonals equals sum of products of opposite sides
- $A C \cdot B D = A B \cdot C D + A D \cdot BC$

Area Formula

Area can be found using a formula similar to Heron’s, based on side lengths only

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.

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.