View source for Power of a point theorem

=STILL WORKING (PLEASE DON'T EDIT YET)=

==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>

<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 <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((1.5,2.598));
label("A",(1.8,2.828));
label("B",(-6,1.4));
dot((-6,1));
label("C",(2.35,-2.4));
dot((2.12,-2.123));
draw((2.3,-1.926)---(-2.12,2.123));
</asy>

=====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>

===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>.

==Proof==

==Problems==

====Introductory (AMC 10, 12)====

====Intermediate (AIME)====

====Olympiad (USAJMO, USAMO, IMO)====