|
|
| (62 intermediate revisions by 2 users not shown) |
| Line 1: |
Line 1: |
| − | ==Theorem:==
| + | #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>
| |
| − | | |
| − | <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));
| |
| − | </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>
| |
| − | | |
| − | =====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)====
| |
