Difference between revisions of "Mass points"
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==Problems== | ==Problems== | ||
[http://www.artofproblemsolving.com/Wiki/index.php/2004_AMC_10B_Problems/Problem_20 2004 AMC 10B #20] | [http://www.artofproblemsolving.com/Wiki/index.php/2004_AMC_10B_Problems/Problem_20 2004 AMC 10B #20] | ||
| − | [http://www.artofproblemsolving.com/Wiki/index.php/2009_AIME_I_Problems/Problem_4] | + | [http://www.artofproblemsolving.com/Wiki/index.php/2009_AIME_I_Problems/Problem_4 2009 AIME I #4] |
PLEASE ADD MORE! | PLEASE ADD MORE! | ||
Revision as of 15:42, 1 January 2014
Mass points is a technique in Euclidean geometry that can greatly simplify the proofs of many theorems concerning polygons, and is helpful in solving complex geometry problems involving lengths. In essence, it involves using a local coordinate system to identify points by the ratios into which they divide line segments. Mass points are generalized by barycentric coordinates.
Mass point geometry involves systematically assigning 'weights' to points using ratios of lengths relating vertices, which can then be used to deduce other lengths, using the fact that the lengths must be inversly proportional to their weight (just like a balanced lever). Additionally, the point dividing the line has a mass equal to the sum of the weights on either end of the line (like the fulcrum of a lever).
Examples
Consider a triangle 
with its three medians drawn, with the intersection points being 
corresponding to 
and 
respectively. Thus, if we label point 
with a weight of 
, 
must also have a weight of since and are equidistant to 
. By the same process, we find 
must also have a weight of 1. Now, since and both have a weight of , must have a weight of 
(as is true for 
and 
). Thus, if we label the centroid 
, we can deduce that 
is 
- the inverse ratio of their weights.
Problems
2004 AMC 10B #20 2009 AIME I #4
PLEASE ADD MORE!
External links
- http://mathcircle.berkeley.edu/archivedocs/2007_2008/lectures/0708lecturesps/MassPointsBMC07.ps
- http://mathcircle.berkeley.edu/archivedocs/1999_2000/lectures/9900lecturespdf/mpgeo.pdf
This article is a stub. Help us out by expanding it.
