# Classical Geometry: Part 3

Here we consider a number of theorems concerned with the Euclidean geometry of the circle. We begin by reviewing some important structures related to the circle.

#### Further Circle Anatomy

The vital description of the circle in terms of its radius and centre may be found in The Shape of Things. The arc, a sometimes underestimated structure, was also presented there.  Here we present further geometric structures needed to develop various circle theorems.

Consider two distinct points, A and B, on the circumference of a circle. When these are joined by the shortest path, the resulting straight line segment is called a chord.

The region enclosed by arc AB and chord AB is known as a segment.

If a chord is extended to infinity in both directions, the resulting line is known as a secant.

If a secant is shifted away from the circle centre, the two points of intersection will move toward each other. When they meet at a single point on the circumference, the line is now called a tangent. We also say that this line is tangent to the circle.

Consider again two distinct points, A and B, on the circumference of a circle. Let O be the centre of the circle. If A and B are joined to O, a pie-shaped figure, AOB, is formed. This figure is called a sector.

We can very reasonably view radii OA and OB as the rays of angle AOB, the sector angle.

#### Angle-at-Centre Theorem

Consider minor arc AB, sector AOB, and its acute angle, ∠AOB , in the following diagram.

We will now show an important result involving ∠AOB and a distinct point P lying on major arc AB.

The result, called the angle-at-centre theorem, is:

∠AOB = 2∠APB .

Proof

Angle in a semi-circle

Angles in the same segment

### Proof of Alternate Segment Theorem

the Tangent-Radius theorem may be combined with the Angle-at-Centre theorem to derive the Alternate Segment theorem.