Difference between revisions of "Power of a point theorem"

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==Theorem:==
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#REDIRECT [[Power of a Point 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 <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>
 
 
 
===Case 2 (Outside the Circle):===
 
 
 
=====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>.
 
 
 
=====Tangent Line=====
 
 
 
====Normal Configuration====
 
 
 
====Tangent Line====
 
 
 
===Case 3 (On the Border/Useless Case):===
 
 
 
**Still working
 
 
 
==Proof==
 
 
 
==Problems==
 
 
 
====Introductory (AMC 10, 12)====
 
 
 
====Intermediate (AIME)====
 
 
 
====Olympiad (USAJMO, USAMO, IMO)====
 

Latest revision as of 22:56, 22 March 2025