Difference between revisions of "Power of a point theorem"
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| − | + | ==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 | + | ===Case 1 (Inside the Circle):=== |
| − | Case 3 (On the Border/Useless Case): | + | 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):=== | ||
| + | |||
| + | =====Classic Configuration===== | ||
| + | |||
| + | =====Tangent Line===== | ||
| + | |||
| + | ====Normal Configuration==== | ||
| + | |||
| + | ====Tangent Line==== | ||
| + | |||
| + | ===Case 3 (On the Border/Useless Case):=== | ||
| + | |||
| + | **Still working | ||
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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)==== | ||
Revision as of 14:01, 23 April 2024
Contents
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 
and 
intersect at a point 
within a circle, then 
Case 2 (Outside the Circle):
Classic Configuration
Tangent Line
Normal Configuration
Tangent Line
Case 3 (On the Border/Useless Case):
- Still working
