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Difference between revisions of "Power of a point theorem"

(→‎Case 1 (Inside the Circle):)
(→‎Classic Configuration)
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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>
 
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>
 +
 +
<asy> draw(circle((0,0),3));
 +
dot((-2.82,1));
 +
label("A",(1.5,2.598));
 +
dot((1.8,2.828));
 +
label("B",(-6,1));
 +
dot((-6,1.4));
 +
label("C",(-6,1.7));
 +
dot((-2.12,2.123));
 +
draw((2.3,-1.926)---(-2.12,2.123));
 +
</asy>
  
 
=====Tangent Line=====
 
=====Tangent Line=====

Revision as of 18:10, 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((-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",(-1.656,1.202)); [/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 $BF\cdot BA=BG\cdot BC$

[asy] draw(circle((0,0),3)); dot((-2.82,1)); label("A",(1.5,2.598)); dot((1.8,2.828)); label("B",(-6,1)); dot((-6,1.4)); label("C",(-6,1.7)); dot((-2.12,2.123)); draw((2.3,-1.926)---(-2.12,2.123)); [/asy]

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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