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.

The chord PAQ

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.

The angle PAQ inside the circle with diameter PQ

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}.

The angle PAQ inside the circle with chord 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}.

The angle PCQ is twice the angle PAQ