Cyclic Polygons

A cyclic polygon is a polygon that has a circumcircle. For example, all triangles are cyclic. All regular polygons are also cyclic because they have a radius; the distance between the center of the polygon and its vertices are equal. Cyclic polygons, especially quadrilaterals, have interesting properties, such as the angles from opposite vertices add up to $180^{\circ}$.

A cyclic polygon is a polygon for which we can circumscribe a circle; i.e. there exists a circle such that all the vertices of the polygon are on the circle. For example, this pentagon here:

We say that pentagon $ABCDE$ is cylic. Great. This definition seems quite simple $-$ in fact, it is. However, the properties of many cyclic polygons turn out to be remarkable, and in future, we will see some examples of these, especially with quadrilaterals. But before we dive into those, let's just try to get some intuition.

1. Which triangles are cyclic?

You guessed it; it turns out that all triangles are cyclic; we can always find a circumcircle for a triangle. In fact, that's why the word circumcircle is defined for any triangle $-$ because it always exists!

2. What are some special cyclic polygons?

It can be proven that all regular polygons are cyclic. The intuition is also quite nice. Let us say that a regular polygon has $n$ sides. The obvious thing is $-$ by golly, we have symmetry! So we can just evenly space $n$ points around a circle and the polygon with those $n$ points as vertices is a regular polygon with $n$-sides. Bam!