What is a cyclic quadrilateral?

A cyclic quadrilateral is a four-sided polygon whose vertices all lie on a common circle. In other words, it is a closed figure that can be made by tracing a path around a circle. The term "cyclic" means "having the property of being recurrent or periodic." A good way to remember this definition is that the prefix "cycle" comes from the Greek word for circle, which is kyklos. So, a cyclic quadrilateral is basically a four-sided figure whose corners all lie on some circle or other.

Cyclic quadrilaterals have some interesting and important properties that make them useful in mathematical problems. For instance, because all the vertices of a cyclic quadrilateral lie on the same circle, we can infer that the opposite sides of the quadrilateral are parallel to each other. This fact can be very helpful when solving certain types of geometry problems.

The properties of cyclic quadrilaterals can be summarized as follows:

1. All four vertices lie on a common circle.
2. Opposite sides of a cyclic quadrilateral are parallel to each other.
3. The diagonals of a cyclic quadrilateral intersect at two points, which are equidistant from the center of the circle.
4. The sum of the angles of a cyclic quadrilateral is 360 degrees.
5. The altitude (or height) from any vertex to the opposite side intersects that side at its midpoint.
6. A cyclic quadrilateral has two pairs of congruent sides if and only if it is an inscribed rectangle (a rectangle whose vertices all lie on the circumference of a circle).
7. A cyclic quadrilateral has two pairs of opposite angles congruent if and only if it is an inscribed square (a square whose vertices all lie on the circumference of a circle).

In conclusion, we have seen that a cyclic quadrilateral is simply a four-sided figure whose vertices all lie on some circle or other. We have also seen that cyclic quadrilaterals have some interesting and important properties, which make them useful in mathematical problems. Thanks for reading!