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] | ||
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[http://www.artofproblemsolving.com/Wiki/index.php/2009_AIME_I_Problems/Problem_4 2009 AIME I #4] | [http://www.artofproblemsolving.com/Wiki/index.php/2009_AIME_I_Problems/Problem_4 2009 AIME I #4] | ||
Revision as of 16:43, 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
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.