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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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| − | ===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] 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 </math>.
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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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