Difference between revisions of "Power of a Point Theorem"
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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]]. | 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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== Theorem == | == Theorem == | ||
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There are three possibilities as displayed in the figures below. | There are three possibilities as displayed in the figures below. | ||
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# Both lines are [[secant line|secants]] of the circle and intersect outside of it (figure on the right). In this case, we have <math> CB\cdot CA = CD\cdot CE. </math> | # Both lines are [[secant line|secants]] of the circle and intersect outside of it (figure on the right). In this case, we have <math> CB\cdot CA = CD\cdot CE. </math> | ||
− | + | [[Image:Pop.PNG|center]] | |
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=== Alternate Formulation === | === Alternate Formulation === | ||
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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 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 | 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 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 | ||
+ | <cmath> | ||
+ | PX^2=PA_1\cdot PB_1=PA_2\cdot PB_2=\cdots=PA_i\cdot PB_i | ||
+ | </cmath> | ||
+ | [[Image:Popalt.PNG|center]] | ||
− | + | 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. | |
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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. | ||
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== Additional Notes == | == Additional Notes == | ||
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The theorem generalizes to higher dimensions, as follows. | The theorem generalizes to higher dimensions, as follows. | ||
− | Let <math> | + | Let <math>P</math> be a point, and let <math>S</math> be an <math>n</math>-sphere. Let two arbitrary lines passing through <math>P </math> intersect <math>S </math> at <math>A_1 , B_1 ; A_2 , B_2 </math>, respectively. Then |
+ | <cmath> | ||
+ | PA_1 \cdot PB_1 = PA_2 \cdot PB_2 | ||
+ | </cmath> | ||
− | + | ''Proof.'' We have already proven the theorem for a 1-sphere (i.e., a circle), so it only remains to prove the theorem for more dimensions. Consider the [[plane]] <math>p </math> containing both of the lines passing through <math>P </math>. The intersection of <math>P </math> and <math>S </math> must be a circle. If we consider the lines and <math>P </math> with respect simply to that circle, then we have reduced our claim to the case of two dimensions, in which we know the theorem holds. | |
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− | ''Proof.'' We have already proven the theorem for a 1-sphere (i.e., a circle), so it only remains to prove the theorem for more dimensions. Consider the [[plane]] <math> | ||
== Problems == | == Problems == | ||
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Find the value of <math> x </math> in the following diagram: | Find the value of <math> x </math> in the following diagram: | ||
− | + | [[Image:popprob1.PNG|center]] | |
[[Power of a Point Theorem/Introductory_Problem_1|Solution]] | [[Power of a Point Theorem/Introductory_Problem_1|Solution]] | ||
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Find the value of <math> x </math> in the following diagram. | Find the value of <math> x </math> in the following diagram. | ||
− | + | [[Image:popprob2.PNG|center]] | |
[[Power of a Point Theorem/Introductory_Problem_2|Solution]] | [[Power of a Point Theorem/Introductory_Problem_2|Solution]] | ||
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([[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> | ([[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> | ||
− | + | [[Image:popprob3.PNG|center]] | |
[[Power of a Point Theorem/Introductory_Problem_3|Solution]] | [[Power of a Point Theorem/Introductory_Problem_3|Solution]] | ||
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==== Other Intermediate Example Problems ==== | ==== Other Intermediate Example Problems ==== | ||
* [[1971_Canadian_MO_Problems/Problem_1 | 1971 Canadian Mathematics Olympiad Problem 1]] | * [[1971_Canadian_MO_Problems/Problem_1 | 1971 Canadian Mathematics Olympiad Problem 1]] | ||
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==See also== | ==See also== | ||
* [[Geometry]] | * [[Geometry]] | ||
* [[Planar figures]] | * [[Planar figures]] | ||
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+ | [[Category:Geometry]] | ||
+ | [[Category:Theorems]] |
Revision as of 11:12, 27 April 2008
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.
Contents
Theorem
There are three possibilities as displayed in the figures 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 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.
The theorem generalizes to higher dimensions, as follows.
Let be a point, and let
be an
-sphere. Let two arbitrary lines passing through
intersect
at
, respectively. Then
Proof. We have already proven the theorem for a 1-sphere (i.e., a circle), so it only remains to prove the theorem for more dimensions. Consider the plane containing both of the lines passing through
. The intersection of
and
must be a circle. If we consider the lines and
with respect simply to that circle, then we have reduced our claim to the case of two dimensions, in which we know the theorem holds.
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
.