Difference between revisions of "Mass points"
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'''Mass points''' is a method in [[Euclidean geometry]] that can greatly simplify the proofs of many theorems. In essence, it involves using a local [[coordinate system]] to identify [[point]]s by the [[ratio]]s into which they divide [[line segment]]s. Mass points are generalized by [[barycentric coordinates]]. | '''Mass points''' is a method in [[Euclidean geometry]] that can greatly simplify the proofs of many theorems. In essence, it involves using a local [[coordinate system]] to identify [[point]]s by the [[ratio]]s into which they divide [[line segment]]s. Mass points are generalized by [[barycentric coordinates]]. | ||
| − | Mass point geometry involves systematically assigning 'weights' to points, which can then be used to deduce 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 | + | Mass point geometry involves systematically assigning 'weights' to points, which can then be used to deduce 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 == | == Examples == | ||
Revision as of 13:18, 14 March 2011
Mass points is a method in Euclidean geometry that can greatly simplify the proofs of many theorems. 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, which can then be used to deduce 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.
External links
- http://mathcircle.berkeley.edu/archivedocs/2007_2008/lectures/0708lecturesps/MassPointsBMC07.ps
- http://mathcircle.berkeley.edu/archivedocs/1999_2000/lectures/9900lecturespdf/mpgeo.pdf
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