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Difference between revisions of "Power of a point theorem"

(→‎Case 2 (Outside the Circle):)
(→‎Case 1 (Inside the Circle):)
(40 intermediate revisions by the same user not shown)
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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),3));
 +
dot((0,0));
 +
dot((-2.82, 1));
 +
label("A",(-3.05,1.25));
 +
dot((1,2.828));
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label("B",(1.25,3.05));
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dot((2.3,-1.926));
 +
label("C",(2.55,-2.346));
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dot((-2.12,2.123));
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label("D",(-2.37,2.507));
 +
</asy>
  
 
===Case 2 (Outside the Circle):===
 
===Case 2 (Outside the Circle):===
  
−
[b]Classic Configuration[/b]
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=====Classic Configuration=====
  
−
**Tangent Line**
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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>
  
−
====Normal Configuration====
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=====Tangent Line=====
  
−
====Tangent Line====
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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):===
  
−
**Still working
+
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==
 +
 
 +
==Problems==
 +
 
 +
====Introductory (AMC 10, 12)====
 +
 
 +
====Intermediate (AIME)====
 +
 
 +
====Olympiad (USAJMO, USAMO, IMO)====

Revision as of 17:54, 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),3)); dot((0,0)); dot((-2.82, 1)); label("A",(-3.05,1.25)); dot((1,2.828)); label("B",(1.25,3.05)); dot((2.3,-1.926)); label("C",(2.55,-2.346)); dot((-2.12,2.123)); label("D",(-2.37,2.507)); [/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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