Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "Power of a point theorem"

(→‎Tangent Line)
Line 6: Line 6:
  
 
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>
  
 
===Case 2 (Outside the Circle):===
 
===Case 2 (Outside the Circle):===
Line 11: 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 </math>.
+
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 $.
  
 
==Proof==
 
==Proof==

Revision as of 17:37, 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$ (Error compiling LaTeX. Unknown error_msg) 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$ (Error compiling LaTeX. Unknown error_msg) AB $and$ AC $with$ AC $[[tangent line|tangent]] to the related circle at$ 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,$ (Error compiling LaTeX. Unknown error_msg) 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)

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=Power_of_a_point_theorem&oldid=218532"