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| − | =STILL WORKING (PLEASE DON'T EDIT YET)=
| + | #REDIRECT [[Power of a Point Theorem]] |
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| − | ==Theorem:==
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| − | 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. Can be useful with [[cyclic quadrilaterals]] as well however with a slightly different application.
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| − | ===Case 1 (Inside the Circle):===
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| − | 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>
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| − | <asy> draw(circle((0,0),3));
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| − | dot((-2.82,1));
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| − | label("A",(-3.05,1.25));
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| − | dot((1,2.828));
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| − | label("B",(1.25,3.05));
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| − | draw((-2.82,1)---(1,2.828));
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| − | dot((2.3,-1.926));
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| − | label("C",(2.55,-2.346));
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| − | dot((-2.12,2.123));
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| − | label("D",(-2.37,2.507));
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| − | draw((2.3,-1.926)---(-2.12,2.123));
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| − | dot((-1.556,1.602));
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| − | label("P",(-1.656,1.202));
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| − | </asy>
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| − | | |
| − | ===Case 2 (Outside the Circle):===
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| − | =====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>
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| − | <asy> draw(circle((0,0),3));
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| − | dot((1.5,2.598));
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| − | label("A",(2,3));
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| − | label("B",(-6,1.6));
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| − | dot((-6,1));
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| − | label("C",(2.55,-2.5));
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| − | dot((2.12,-2.123));
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| − | dot((-2.996,-0.155));
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| − | label("G",(-3.350, -0.6));
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| − | dot((-2.429,1.761));
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| − | label("F",(-2.729,2.061));
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| − | draw((1.5,2.598)---(-6,1));
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| − | draw((2.12,-2.123)---(-6,1));
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| − | </asy>
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| − | =====Tangent Line=====
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| − | 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>
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| − | <asy> draw(circle((0,0),3));
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| − | dot((0,3));
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| − | label("C",(0,3.5));
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| − | dot((-8,3));
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| − | label("A",(-8,3.5));
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| − | dot((2.5,-1.658));
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| − | label("B",(2.8,-1.958));
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| − | draw((0,3)---(-8,3));
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| − | draw((2.5,-1.658)---(-8,3));
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| − | dot((-2.907,0.741));
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| − | label("D",(-3.357,0.421));
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| − | </asy>
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| − | | |
| − | ===Case 3 (On the Border/Useless Case):===
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| − | If two chords, <math> AB </math> and <math> AC </math>, have <math> A </math> 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>.
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| − | <asy> draw(circle((0,0),3));
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| − | dot((1,2.828));
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| − | label("A",(1.4,3.028));
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| − | dot((-2.5,-1.658));
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| − | label("B",(-2.8,-1.958));
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| − | dot((2.04,-2.2));
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| − | label("C",(2.34,-2.5));
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| − | draw((1,2.828)---(-2.5,-1.658));
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| − | draw((1,2.828)---(2.04,-2.2));
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| − | </asy>
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| − | | |
| − | ==Proof==
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| − | Yo, who ever is helping edit rn, want to start working on the proof for each case? I think it is similar triangle based and relatively easy. Also, drop your user at the bottom because I want you to get credits.
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| − | ==Problems==
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| − | ====Introductory (AMC 10, 12)====
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| − | ====Intermediate (AIME)====
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| − | ====Olympiad (USAJMO, USAMO, IMO)====
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