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| − | ==Theorem:==
| + | #REDIRECT [[Power of a Point 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.
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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),5)); </asy> <math>
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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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| − | =====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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| − | ===Case 3 (On the Border/Useless Case):===
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| − | 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 $.
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| − | ==Proof==
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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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