Chapter 1

The Power of a Point

and

Elliptic and Hyperbolic Families of Circles

Chapter 0 has prepared us to begin the Jouney Into Geometries. The material in this chapter recasts the discoveries of Alice in Chapter 1 of the text by Sved into a more formal structure.

The "power of a point" is the key concept in this chapter. The formal definition is:

Definition:

Given a circle C and a point P, we define the power of P with respect to C, written P(C), by
P(C) = d2 - r2

where r is the radius of the circle and d is the distance from P to the center of the circle.

 This quantity gives us an idea of the relation of a given point to a given circle. The power of the point is zero if the point lies on the circle (d=r), positive if the point lies outside the circle (dr) and negative if the point lies inside the circle (d<r).

While the power of a point may seem more complicated than just using the distance from P to the circle, it turns out that there are several equivalent geometric ways of computing the power of a point and that these provide important tools when we look at the relation between the circles in elliptic and hyperbolic families at the end of the chapter.

The first part of the chapter discusses the various ways the power of a point can be computed and the later part defines and discusses the relationships between an elliptic family of circles and corresponding hyperbolic family of circles.

Alice discovers two equivalent ways of finding the power of a point when the point lies outside the circle and then a relation that holds regardless of whether the point is in or out of the circle.

Her first discovery is:

Theorem: (Constant Chord Product)

If two chords of a circle, AB and CD, when extended, meet at point P, then: (PA) (PB) = (PC) (PD).

Proof:

In other words: if any line is drawn from P to meet the circle in a chord, the product of the distances from P to the end points of the chord intercepted by the line does not depend on the chord intercepted. The product is always the same for any such chord. The next two theorems show that this constant product is the power of P with respect to the circle.

Theorem: (Tangent-Chord Equivalence)

If P is exterior to a circle and T is on the circle with PT tangent to the circle and if AB is a chord of the circle that, when produced, passes through P, then PT2 = (PA) (PB).

Proof:

Theorem: (Tangent Power Formula)

Let P be exterior to a circle of radius r, PT tangent to the circle at T, and d be the distance from P to the center of the circle, then d2-r2=PT2. So the power of P with respect to the circle is equal to PT2.

Proof:

The relations in the above three theorems are illustrated in the following picture:

An obvious corollary to the above three theorems is that the quantities, P(C) = d2 - r2 = PT2 = (PA) (PB) in the above theorems are all equal to the power of P, when P is external to the circle.

The following result is important because it provides a way to compute the power of a point w. r. t. a circle regardless of whether the point is inside or outside the circle.

Theorem: (Pythagorean Power Formula)

Let P be any point and C be any circle. Let j be any line through the center of C that misses P and let E be a point on j. Then P(C)=E(C)+h2, where h=PE, iff PE is perpendicular to j.

 Proof:

We must show that:

1. If P is any point and C any circle and j any line through the center of C that misses P and E is the foot of the perpendicular from P to j, then P(C)=E(C) + h2, where h=PE.
and
2. If P is any point and C any circle and j any line through the center of C that misses P and E is a point on j such that P(C)=E(C) + h2, where h=PE, then PE is perpendicular to j.
The next notion introduced is that of an elliptic family of circles.

Definition:

Let line segment AB be given. The set of all circles which have AB as a chord is called an elliptic family of circles with common chord AB. One can similarly define a parabolic family of circles which will be used later in the text.

Definition:

Let point P on line segment AB be given. The set of all circles which are tangent to AB at P is called a parabolic family of circles. Note that the centers of an elliptic family of circles line up as do the centers of a parabolic family:

Theorem: (Elliptic Centers)

If an elliptic family of circles share the common chord AB, then the centers of the circles all lie on the perpendicular bisector of AB.

Proof:

Theorem: (Parabolic Centers)

If a parabolic family of circles are tangent to line AB at point P, then the centers of the circles all lie on the perpendicular to AB through P.

Proof:

We now pose the question: Given two circles, when does a point have the same power w.r.t. both circles?

Definition:

Given two circles, the set of all points P such that P has the same power with respect to each circle is called the radical axis of the circles.

Theorem: (Fundamental Radical Axis Theorem)

Let E be a point on the line, j, which joins the centers of two circles. E has the same power w.r.t. both circles if and only if each point on the line through E perpendicular to j has the same power with respect to both circles.

Proof:

Note: 1. To show that the radical axis the two circles is the line through E perpendicular to j, we must show that there is exactly one point E on the line through the centers of the two circles that has the same power with respect to each circle.

2. The theorem says nothing about whether the circles intersect or not, so its proof must not make any assumptions about intersections.

 Theorem: (Unique Radical Axis Theorem)
Let C1 and C2 be circles with distinct centers O1 and O2. Then there is a unique point E on the line through O1 and O2 such that E(O1)=E(O2).
Proof:

The next two theorems show that:

1. the radical axis of any two circles in an elliptic family is the line through the common chord and that
2. the radical axis of any two circles in a parabolic family is the line through the common tangent.
First we make the definition:

Definition:

If every pair of circles in a family of circles has the same radical axis, we say the family is coaxial and the common radical axis is the radical axis of the family.

Theorem: (Radical Axis of Tangent Circles)

If two circles are tangent at point P, then their radical axis is the line through P that is tangent to each circle at P.

Proof:

Corollary:(Parabolic Family Radical Axis)

The radical axis of a parabolic family of circles is the common tangent line of the family.

Theorem: (Radical Axis of Common Chord Circles)

If two circles share a common chord, then their radical axis is the line through the common chord.

Proof:

Corollary:(Elliptic Family Radical Axis)

The radical axis of an elliptic family of circles is the line through the common chord of the family. There is another important family of circles that Alice encounters: the hyperbolic family. It turns out that hyperbolic families are coaxial and the radical axis is perpendicular to a line containing the centers of all the circles in the family, just as in the case of parabolic and elliptical families. Indeed, each hyperbolic family is "dual" to an elliptical family in the sense that all centers of the hyperbolic family lie on the radical axis of the elliptic family and all centers of the elliptical family lie on the radical axis of the hyperbolic family. This will become clear below for our next job is to explicitly describe a hyperbolic family of circles and the radical axis of such a family. We begin by defining orthogonal circles.

Definition:

Two circles are called orthogonal if they intersect and at each point of intersection the tangent line to each circle passes through the center of the other circle. Note: The definition implies that the tangents to the circles at the points of intersection are perpendicular to each other.

The next theorem shows how to construct a circle orthogonal to every circle in an elliptic family.

Theorem: (Hyperbolic Family Construction)

Let P be a point on the radical axis of an elliptic family that has positive power w.r.t. the family. Given any circle in the elliptic family, draw PT so that PT is tangent to the circle. The set of all such points T lie on a circle, C, with center P which is orthogonal to each circle in the elliptic family.

Proof:

We can now define a hyperbolic family:

Definition:

The family of circles with centers, P, such that P is on the radical axis of an elliptic family and which are orthogonal to each member of the elliptic family as described in the above theorem is called the hyperbolic family corresponding to the elliptic family. Clearly the centers of a hyperbolic family all lie on the radical axis of an elliptic family.

We can also find the radical axis of the hyperbolic family:

Theorem: (Hyperbolic Family Radical Axis)

The line containing the centers of an elliptic family of circles is the radical axis of the corresponding hyperbolic family of circles.

Proof:

This concludes the companion to the journey in Chapter 1.

Chapter 2.
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