CIRCLE RELATIONSHIPS IN BASIC PLANE GEOMETRY — AND AN EXTENSION

As we begin our exploration of the circle, the one aspect that most people recall is that the Greek letter p (pi) is somehow related to this important geometric shape. Most will recall that p represents the ratio of the circumference of a circle to its diameter. The two formulas that come instantly to mind are that the circumference of the circle is equal to 2pr, and that the area of the circle is equal to pr. In each case, r is the length of the radius of the circle. However, before we inspect the circle for its multitude of applications and appearances, we will review some of the basic essentials that you may have forgotten from your high-school geometry course.

First, we will review some basic terminology related to circles. We all know that the radius is the line segment joining the center of the circle to any point on the circle, while the diameter is the line segment joining two points on the circle and that also contains the center of the circle. A line segment joining two points on the circle is called a chord. A line is said to be tangent to the circle if it touches a circle at exactly one point; and a line that intersects the circle in two points is called a secant. Two more definitions to review: the area within a circle bounded by two radii and the arc between them is called a sector, and the area within a circle bounded by a chord and the arc of the circle is called a segment of the circle.

Some relationships useful to our study of the circle are the following:

• Any three non-collinear points determine a unique circle.

* The non-collinear points A, B, and C determine the unique circle O. (See figure 1.1.)

• The perpendicular bisector of a chord contains the center circle and the midpoints of the arcs it intersects.

* Line CD is the perpendicular bisector of AB; therefore, line CD contains the center of the circle and points C and D are the midpoints of arcs AB. (See figure 1.2.)

• A line perpendicular to a radius at the endpoint on the circle is tangent to the circle.

* Line AB is perpendicular to radius OC at point C and is therefore tangent to the circle at point C. (See figure 1.3.)

• From a point outside the circle, two tangents drawn to the circle have equal segments to the points of tendency.

* Tangents PA and PB are equal in length. (See figure 1.4.)

• A polygon is said to be inscribed in a circle if all of its vertices are on the circle.

* Polygon ABCDE is inscribed in circle O. (See figure 1.5.)

• A circle is said to be inscribed in a polygon if all of the polygon's sides are tangent to the circle. (See figure 1.6.)

* Circle O is inscribed in polygon ABCDE.

• If a secant segment and a tangent segment to the same circle share an endpoint in the exterior of the circle, then the length of the tangent segment is the mean proportional between the length of the secant segment and the length of its external segment.

* Tangent AP is the mean proportional between PC and PB, so that

PC/AP = AP/PB

(see figure 1.7).

• If two secant segments of the same circle share an endpoint in the exterior of the circle, then the product of the lengths of one secant segment and its external segment equals a product of the lengths of the other secant segment and its external segment.

* For the two secants PED and PBC, the following is true:

PD · PE = PC · PB. (See figure 1.8.)

• If two chords intersect in the interior of a circle, the product of the lengths of the segments of one chord equals the product of the lengths of the segments of the other chord.

* For the two chords intersecting at the point P, the following is true: PA · PB = PC · PD. (See figure 1.9.)

• A central angle is one formed by two radii of a circle, and it is equal in measure to its intercepted arc.

* The measure of angle AOB is equal to the degree measure of arc AB. (See figure 1.10.)

• The measure of an inscribed angle, one formed by two chords intersecting on the circle, is one half the measure of its intercepted arc.

* The measure of angle APB is one-half the measure of arc AB. (See figure 1.11.)

• The measure of an angle formed by a tangent and a chord of a circle is one half the measure of its intercepted arc.

* The measure of angle ABP is one-half the measure of arc AB. (See figure 1.12.)

• The measure of an angle formed by two chords intersecting at a point in the interior of the circle is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle.

* The measure of angle BPD is equal to one-half the measure of the sum of the measures of arcs BD and AC. (See figure 1.9.)

• The measure of an angle formed by two secants of the circle intersecting at a point in the exterior of the circle is equal to one-half the difference of the measures of the intercepted arcs.

* The measure of angle DPC is one-half the measure of arc DC minus arc EB. (See figure 1.8.)

• The measure of an angle formed by a secant and a tangent to a circle, which intersect at a point in the exterior of the circle, is equal to one-half the difference of the measures of the intercepted arcs.

* The measure of angle APC is equal to one-half the measure of arc AC minus arc AB. (See figure 1.7.)

• The measure an angle formed by two tangents from a common external point to a circle is equal to one half the difference of the measures of the intercepted arcs. We also notice that the measure of the angle formed by two tangents from an external point is supplementary to the measure of the closer intercepted arc.

* The measure of angle APB is equal to the difference of the two arcs AB. Additionally, the measure of angle APB is supplementary to the closer arc AB. (See figure 1.4.)

This brief review of the essentials of circle relationships that are introduced in the high-school geometry course should...