Mass points

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 Problems/Problem 20

2009 AIME I Problems/Problem 4

2001 AIME I Problems/Problem 7

1992 AIME Problems/Problem 14

1988 AIME Problems/Problem 12

1989 AIME Problems/Problem 15

External links

• http://mathcircle.berkeley.edu/archivedocs/2007_2008/lectures/0708lecturespdf/MassPointsBMC07.pdf
• http://mathcircle.berkeley.edu/archivedocs/1999_2000/lectures/9900lecturespdf/mpgeo.pdf

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