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Difference between revisions of "Power of a point theorem"
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If two chords <math> AB </math> and <math> CD </math> intersect at a point <math> P </math> within a circle, then <math> AP\cdot BP=CP\cdot DP </math>
 
If two chords <math> AB </math> and <math> CD </math> intersect at a point <math> P </math> within a circle, then <math> AP\cdot BP=CP\cdot DP </math>
  
−
$ <asy> draw(circle((0,0),5)); </asy> <math>
+
<asy> draw(circle((0,0),5)); </asy>
  
 
===Case 2 (Outside the Circle):===
 
===Case 2 (Outside the Circle):===
Line 13: Line 13:
 
=====Classic Configuration=====
 
=====Classic Configuration=====
  
−
Given lines </math> AB <math> and </math> CB <math> originate from two unique points on the [[circumference]] of a circle (</math> A <math> and </math> C <math>), intersect each other at point </math> B <math>, outside the circle, and re-intersect the circle at points </math> F <math> and </math> G <math> respectively, then </math> BF\cdot BA=BG\cdot BC <math>
+
Given lines <math> AB </math> and <math> CB </math> originate from two unique points on the [[circumference]] of a circle (<math> A </math> and <math> C </math>), intersect each other at point <math> B </math>, outside the circle, and re-intersect the circle at points <math> F </math> and <math> G </math> respectively, then <math> BF\cdot BA=BG\cdot BC </math>
  
 
=====Tangent Line=====
 
=====Tangent Line=====
  
−
Given Lines </math> AB <math> and </math> AC <math> with </math> AC <math> [[tangent line|tangent]] to the related circle at </math> C <math>, </math> A <math> lies outside the circle, and Line </math> AB <math> intersects the circle between </math> A <math> and </math> B <math> at </math> D <math>, </math> AD\cdot AB=AC^{2} <math>
+
Given Lines <math> AB </math> and <math> AC </math> with <math> AC </math> [[tangent line|tangent]] to the related circle at <math> C </math>, <math> A </math> lies outside the circle, and Line <math> AB </math> intersects the circle between <math> A </math> and <math> B </math> at <math> D </math>, <math> AD\cdot AB=AC^{2} </math>
  
 
===Case 3 (On the Border/Useless Case):===
 
===Case 3 (On the Border/Useless Case):===
  
−
If two chords, </math> AB <math> and </math> AC <math>, 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 </math> 0 <math> so no matter what, the constant product is </math> 0 $.
+
If two chords, <math> AB </math> and <math> AC </math>, 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 <math> 0 </math> so no matter what, the constant product is <math> 0 </math>.
  
 
==Proof==
 
==Proof==

Revision as of 16:38, 23 April 2024

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 $AB$ and $CD$ intersect at a point $P$ within a circle, then $AP\cdot BP=CP\cdot DP$

[asy] draw(circle((0,0),5)); [/asy]

Case 2 (Outside the Circle):

Classic Configuration

Given lines $AB$ and $CB$ originate from two unique points on the circumference of a circle ($A$ and $C$), intersect each other at point $B$, outside the circle, and re-intersect the circle at points $F$ and $G$ respectively, then $BF\cdot BA=BG\cdot BC$

Tangent Line

Given Lines $AB$ and $AC$ with $AC$ tangent to the related circle at $C$, $A$ lies outside the circle, and Line $AB$ intersects the circle between $A$ and $B$ at $D$, $AD\cdot AB=AC^{2}$

Case 3 (On the Border/Useless Case):

If two chords, $AB$ and $AC$, 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 $0$ so no matter what, the constant product is $0$.

Proof

Problems

Introductory (AMC 10, 12)

Intermediate (AIME)

Olympiad (USAJMO, USAMO, IMO)

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