Introduction: A circle is all points equidistant from one point called the center of the circle. Segments drawn within, through, or tangent to a circle create angles which we will now define and measure. Intersecting segments also create smaller segments. We will learn how to relate the lengths of these segments mathematically.

Important facts:

1. The measure of a central angle is the same as the measure of the intercepted arc.
2. The measure of an inscribed angle is half the measure of the intercepted arc.
3. A segment connecting two points on a circle is called a chord.
4. A line passing through two points on a circle is called a secant.
5. A line external to a circle, passing through one point on the circle, is a tangent.

The Lesson:

We show circle O below in figure a. Points A, B, C, and D are on the circle. The segments AP and DP are secants because they intersect the circle in two points. Notice that the arcs intercepted are arcs CB and AD.

How does the measure of angle P relate to the arcs CB and AD?

By drawing the segments DC and AB shown in red, we form the triangles ABP and DCP. These are similar triangles because they have angle P in common and angles A and D must be equal because they are inscribed angles intercepting the same arc, CB. This means that angles A and D must equal one half the measure of arc CB.

In figure a, we also show angle 1 which is angle ACD because we will need to refer to it below. Notice that angle 1 is inscribed and intercepts arc AD. Therefore, angle 1 has measure equal to one half of arc AD.

 ► ANGLE outside a circle formed by two secants:

Below in figure b, we only show triangle DCP from the circle diagram shown in figure a above.

As shown in the diagram above, ÐDCP is supplementary to angle 1.

The three angles of triangle DCP must have a sum of 180°.

Solving this equation for angle P yeilds

This means that the measure of angle P, an angle external to a circle and formed by two secants, is equal to one half the difference of the intercepted arcs.

 ► ANGLE outside a circle formed by secants/tangents:

We just learned that the measure of an external angle P (as shown in figure b) when formed by two secants is equal to one half the difference of the measures of the intercepted arcs.

In a related result, if one (or both) of the segments is tangent, as in segment PC in figure c shown below, the external angle P is also one half the difference of the intercepted arcs DC and CB.

 ► SEGMENTS formed by secants, drawn from a point, intersecting a circle:

Figure a is shown again for reference.

We have already noted that triangles ABP and DCP are similar. This gives corresponding sides as follows:

PC ~ PB and PD ~ PA

In a proportion true for corresponding parts of similar triangles, we have

Notice that these are the products of the exterior part of each secant with each secant's entire length,

 ► SEGMENTS formed by a secant and a tangent, drawn from a point, intersecting a circle:

In the case where one of the segments forming angle P is a tangent, we show figure c again.

We have added segments CB and DC. Looking at triangles PCB and PDC, we have the following:

1. both triangles share angle P and
2. angle D and angle PCB both have measure 1/2 arc CB, the intercepted arc

Thus, triangles PCB and PDC are similar. Since

sides PC ~ PD and
sides PB ~ PC

we can write the proportion

We summarize:

• The angle formed outside a circle by intersecting secants (or a secant and tangent or two tangents) is equal in measure to ½ the difference of the intercepted arcs.
• If two secants meet at a point outside a circle, the product of the exterior part of one secant with its entire length is equal to the product of the exterior part of the other secant with its entire length.
• If a secant and a tangent meet at a point outside a circle, the product of the exterior part of the secant with its entire length is equal to the square of the tangent segment.

Let's Practice:

1. In circle O below, suppose that angle P has measure 25° and arc AD has measure 70°.

What is the measure of arc CB?

We must have

which gives us

2. In circle O below, secants are drawn from point P. PC = 10, PB = 9, AC = x, and DB = 12.

What is the length of secant PA?

Notice that

(PC)(PA) = (PB)(PD)

which is the same as

10(10 + x ) = 9(9 + 12) = 189

which gives us

100 + 10x = 189
10x = 89
x = 8.9

PA = 10 + x
PA = 18.9