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.

CIRCLE_ANATOMY4

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.

CIRC_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.

TANGENT_LN

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.

SECTOR_area

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.

CIRCLE_ANATOMY2

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

ANG_at_Centre

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

∠AOB = 2∠APB .

Proof

Angle in a semi-circle

Angles in the same segment

Tangent-Radius


Equal_Chords_Thm


 

 

 

Proof of Alternate Segment Theorem

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


Cyclic quad …