Difference between revisions of "Power of a Point Theorem"
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([[ARML]]) Chords <math> AB </math> and <math> CD </math> of a given circle are perpendicular to each other and intersect at a right angle. Given that <math> BE = 16, DE = 4, </math> and <math> AD = 5 </math>, find <math> CE </math>. | ([[ARML]]) Chords <math> AB </math> and <math> CD </math> of a given circle are perpendicular to each other and intersect at a right angle. Given that <math> BE = 16, DE = 4, </math> and <math> AD = 5 </math>, find <math> CE </math>. | ||
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=== Intermediate === | === Intermediate === | ||
Revision as of 08:25, 1 July 2006
Contents
Introduction
The Power of a Point Theorem expresses the relation between the lengths involved with the intersection of two lines between each other and their intersections with a circle.
Theorem
There are three possibilities as displayed in the figure below.
- The two lines are secants of the circle and intersect inside the circle (figure on the left). In this case, we have
. - One of the lines is tangent to the circle while the other is a secant (middle figure). In this case, we have
. - Both lines are secants of the circle and intersect outside of it (figure on the right). In this case, we have
.
.
Alternate Formulation
This alternate formulation is much more compact, convenient, and general.
Consider a circle O and a point P in the plane where P is not on the circle. Now draw a line through P that intersects the circle in two places. The power of a point theorem says that the product of the the length from P to the first point of intersection and the length from P to the second point of intersection is constant for any choice of a line through P that intersects the circle. This constant is called the power of point P. For example, in the figure below
Notice how this definition still works if 
and 
coincide (as is the case with X). Consider also when P is inside the circle. The definition still holds in this case.
Additional Notes
One important result of this theorem is that both tangents from a point 
outside of a circle to that circle are equal in length.
Problems
The problems are divided into three categories: introductory, intermediate, and olympiad.
Introductory
Problem 1
Find the value of 
in the following diagram:
Problem 2
Find the value of in the following diagram.
Problem 3
(ARML) In a circle, chords 
and 
intersect at 
. If 
and 
, find the ratio 
Problem 4
(ARML) Chords and of a given circle are perpendicular to each other and intersect at a right angle. Given that 
and 
, find 
.
Intermediate
Problem 1
Two tangents from an external point are drawn to a circle and intersect it at 
and 
. A third tangent meets the circle at 
, and the tangents 
and 
at points 
and , respectively. Find the perimeter of 
.
This page is in need of some relevant examples or practice problems. Help us out by adding some. Thanks.
