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 of two lines | + | == 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 [[secant]]s of the circle and intersect inside the circle (figure on the left). In this case, we have <math> AE\cdot CE = BE\cdot DE </math>. | ||
| + | # One of the lines is [[tangent]] to the circle while the other is a [[secant]] (middle figure). In this case, we have <math> AB^2 = BC\cdot BD </math>. | ||
| + | # Both lines are [[secant]]s 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> | ||
| + | |||
| + | <center>[[Image:Pop.PNG]]</center> | ||
| + | |||
| + | === 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 | ||
| + | |||
| + | <center><math> PX^2 = PA_1\cdot PB_1 = PA_2\cdot PB_2 = \cdots = PA_i\cdot PB_i </math></center> | ||
| + | |||
| + | <center>[[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. | ||
==See also== | ==See also== | ||
* [[Geometry]] | * [[Geometry]] | ||
* [[Planar figures]] | * [[Planar figures]] | ||
| − | |||
Revision as of 17:11, 24 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.
- 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.
