Difference between revisions of "Power of a point theorem"
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draw((2.3,-1.926)---(-2.12,2.123)); | draw((2.3,-1.926)---(-2.12,2.123)); | ||
dot((-1.556,1.602)); | dot((-1.556,1.602)); | ||
| − | label("P",(-1. | + | label("P",(-1.756,1.202)); |
</asy> | </asy> | ||
Revision as of 18:03, 23 April 2024
Contents
Theorem:
There are three unique cases for this theorem. Each case expresses the relationship between the length of line segments that pass through a common point and touch a circle in at least one point.
Case 1 (Inside the Circle):
If two chords 
and 
intersect at a point 
within a circle, then 
![[asy] draw(circle((0,0),3)); dot((-2.82,1)); label("A",(-3.05,1.25)); dot((1,2.828)); label("B",(1.25,3.05)); draw((-2.82,1)---(1,2.828)); dot((2.3,-1.926)); label("C",(2.55,-2.346)); dot((-2.12,2.123)); label("D",(-2.37,2.507)); draw((2.3,-1.926)---(-2.12,2.123)); dot((-1.556,1.602)); label("P",(-1.756,1.202)); [/asy]](http://latex.artofproblemsolving.com/7/b/e/7be3834ffeb94b6c5c2c85b93fbd852b7bf3c0d2.png)
Case 2 (Outside the Circle):
Classic Configuration
Given lines and 
originate from two unique points on the circumference of a circle (
and 
), intersect each other at point 
, outside the circle, and re-intersect the circle at points 
and 
respectively, then 
Tangent Line
Given Lines and 
with tangent to the related circle at , lies outside the circle, and Line intersects the circle between and at 
, 
Case 3 (On the Border/Useless Case):
If two chords, and , have A on the border of the circle, then the same property such that if two lines that intersect and touch a circle, then the product of each of the lines segments is the same. However since the intersection points lies on the border of the circle, one segment of each line is 
so no matter what, the constant product is .
