This is illustrated in the diagram above, left.

The above diagram to the right shows that two

tangents through a common point Poutside the circle

produce line segments PA and PB of equal length.

This follows from the fact that the two triangles pro-

duced are both right triangles of the same height with

a shared hypotenuse, and are hence congruent. Thus:

If PA and PB are tangents to a circle at

points Aand B, respectively, then PA and

PB have the same length.

2. Inscribed-Angle Theorems

In the diagram below, left, angles α(the peripheral

angle) and β(the central angle) are subtended by the

same

ARC

. Thus we have:

For angles subtended by the same arc, the

central angle is always twice that of the

peripheral angle.

This is proved by drawing a radius from the cen-

ter Oto the point at which angle αlies to create two

isosceles triangles. Following the left-hand side of the

next diagram, and noting that the interior angles of a

triangle sum to 180°, we thus have x+ y= αand

(180 – 2x) + (180 – 2y) + β= 360, from which it fol-

lows that β= 2α. A modification of this argument

shows that the result is still true even if the peripheral

angle is located as shown in the right-hand side of

the diagram, or if the arc under consideration is

more than half the

PERIMETER

of the circle.

The next three results follow (see diagram

below, right):

i. All angles inscribed in a circle subtended by the

same arc are equal,

ii. All angles inscribed by a diameter are right

angles. (This is known as the theorem of Thales.)

circle theorems 77

Tangent theorems

Central and peripheral angles

Proving the central-angle/peripheral-angle theorem

Consequences of the central-angle/peripheral-angle theorem