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Difference between revisions of "Power of a Point Theorem"
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== Introduction ==
 
== Introduction ==
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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]].
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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 ==
 
== Theorem ==
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Notice how this definition still works if <math> A_k </math> and <math> B_k </math> coincide (as is the case with X).  Consider also when P is inside the circle.  The definition still holds in this case.
 
Notice how this definition still works if <math> A_k </math> and <math> B_k </math> coincide (as is the case with X).  Consider also when P is inside the circle.  The definition still holds in this case.
 +
 +
== Problems ==
 +
The problems are divided into three categories: introductory, intermediate, and olympiad.
 +
 +
=== Introductory ===
 +
==== Problem 1 ====
 +
Find the value of <math> x </math> in the following diagram:
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<center>[[Image:popprob1.PNG]]</center>
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 +
[[Power of a Point Theorem Introductory Problem 1|Solution]]
 +
 +
==== Problem 2 ==== 
 +
Find the value of <math> x </math> in the following diagram.
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 +
<center>[[Image:popprob2.PNG]]</center>
 +
 +
[[Power of a Point Theorem Introductory Problem 2|Solution]]
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 +
==== Problem 3 ====
 +
([[ARML]]) In a circle, chords <math> AB </math> and <math> CD </math> intersect at <math> R </math>. If <math>AR:BR = 1:4 </math> and <math> CR:DR = 4:9 </math>, find the ratio <math> AB:CD. </math>
 +
 +
<center>[[Image:popprob3.PNG]]</center>
 +
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[[Power of a Point Introductory Problem 3|Solution]]
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 +
==== Problem 4 ====
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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>.
 +
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[[Power of a Point Introductory Problem 4|Solution]]
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{{problems}}
  
 
==See also==
 
==See also==
 
* [[Geometry]]
 
* [[Geometry]]
 
* [[Planar figures]]
 
* [[Planar figures]]

Revision as of 22:18, 30 June 2006

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.

  1. The two lines are secants of the circle and intersect inside the circle (figure on the left). In this case, we have $AE\cdot CE = BE\cdot DE$.
  2. One of the lines is tangent to the circle while the other is a secant (middle figure). In this case, we have $AB^2 = BC\cdot BD$.
  3. Both lines are secants of the circle and intersect outside of it (figure on the right). In this case, we have $CB\cdot CA = CD\cdot CE.$
Pop.PNG

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

$PX^2 = PA_1\cdot PB_1 = PA_2\cdot PB_2 = \cdots = PA_i\cdot PB_i$
Popalt.PNG

Notice how this definition still works if $A_k$ and $B_k$ coincide (as is the case with X). Consider also when P is inside the circle. The definition still holds in this case.

Problems

The problems are divided into three categories: introductory, intermediate, and olympiad.

Introductory

Problem 1

Find the value of $x$ in the following diagram:

Popprob1.PNG

Solution

Problem 2

Find the value of $x$ in the following diagram.

Popprob2.PNG

Solution

Problem 3

(ARML) In a circle, chords $AB$ and $CD$ intersect at $R$. If $AR:BR = 1:4$ and $CR:DR = 4:9$, find the ratio $AB:CD.$

Popprob3.PNG

Solution

Problem 4

(ARML) Chords $AB$ and $CD$ of a given circle are perpendicular to each other and intersect at a right angle. Given that $BE = 16, DE = 4,$ and $AD = 5$, find $CE$.

Solution

This page is in need of some relevant examples or practice problems. Help us out by adding some. Thanks.

See also

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