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]]. | ||
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| + | 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 fulcrom of a lever). | ||
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| + | == Examples == | ||
| + | Consider a triangle <math>ABC</math> with its three [[median]]s drawn, with the intersection points being <math>D, E, F,</math> corresponding to <math>AB, BC,</math> and <math>AC</math> respectively. Thus, if we label point <math>A</math> with a weight of <math>1</math>, <math>B</math> must also have a weight of <math>1</math> since <math>A</math> and <math>B</math> are equidistant to <math>D</math>. By the same process, we find <math>C</math> must also have a weight of 1. Now, since <math>A</math> and <math>B</math> both have a weight of <math>1</math>, <math>D</math> must have a weight of <math>2</math> (as is true for <math>E</math> and <math>F</math>). Thus, if we label the centroid <math>P</math>, we can deduce that <math>DP:PC</math> is <math>1:2</math> - the inverse ratio of their weights. | ||
== External links == | == External links == | ||
Revision as of 18:58, 25 July 2008
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 fulcrom 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
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