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 $P$ and a positive real number $r$, the **circle with center $P$ and radius $r$** consists of all points in the plane of distance $r$ from $P$.

### Circles and Line Segments

A **diameter** of a circle is any line segment with both endpoints on the circle which passes through the center $P$. *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 $P$ 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 $P$ and $Q$ are distinct points on a circle, then we can imagine the chord $\overline{PQ}$ dividing the circle into two parts, called **arcs**. When we use the notation $\stackrel{\frown}{PQ}$ 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 $\stackrel{\frown}{PAQ}$ refers to the upper arc.

### Drawing angles in circles

**Theorem. ** If $\overline{PQ}$ is a diameter of a circle and $A$ is *any* other point on the circle, then the angle $\angle{PAQ}$ is a right angle.

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

**Theorem. **If $\overline{PQ}$ is a chord of a circle and $A$ is *any* other point on the circle, then the measure of $\angle{PAQ}$ does not depend on the choice of $A$ – we call this the **angle subtended by **$\stackrel{\frown}{PQ}$.

This angle is $90^\circ$ if and only if the chord $\overline{PQ}$ 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 $\overline{PQ}$, the angle $\angle{PCQ}$ where $C$ is the center of the circle is called the **central angle subtended by $\overline{PQ}$**.

**Theorem. **The central angle subtended by $\overline{PQ}$ is always twice the angle subtended by $\overline{PQ}$.

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