Circles arise naturally in geometry (they are an immediate consequence of the notion of distance), and we use them extensively in geometric constructions (after all, a compass is essentially a “circle-drawing tool”). However, we focus much more on proving facts about triangles, and figures involving triangles. What can be said about circles themselves?

### Circle

Definition. Given a point and a positive real number , the circle with center and radius consists of all points in the plane of distance from .

### Circles and Line Segments

A diameter of a circle is any line segment with both endpoints on the circle which passes through the center . Note: Sometimes we use the word “diameter” to mean the length of such a segment.

A radius of a circle is any line segment with one endpoint at the center and the other endpoint on the circle. NOTE: Once again, we sometimes use the word “radius” to mean the length of such a segment.

### Chords and Arcs

A chord of a circle is any line segment with both endpoints on the circle.

Arc. If and are distinct points on a circle, then we can imagine the chord dividing the circle into two parts, called arcs. When we use the notation it is usually clear from context which of these two arcs we mean – but sometimes, an additional letter is provided to help the reader.

Example: In the image below refers to the upper arc.

### Drawing angles in circles

Theorem. If is a diameter of a circle and is any other point on the circle, then the angle is a right angle.

What if we use a different chord (not a diameter)?

Theorem. If is a chord of a circle and is any other point on the circle, then the measure of does not depend on the choice of – we call this the angle subtended by .

This angle is if and only if the chord is a diameter.

Finally, what if we consider angles with vertex at the center of the circle (instead of on the circle itself)?

Definition. For a chord , the angle where is the center of the circle is called the central angle subtended by .

Theorem. The central angle subtended by is always twice the angle subtended by .