Mass points

Revision as of 17:56, 12 March 2014 by Flamefoxx99 (talk | contribs) (Problems)

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).


Consider a triangle $ABC$ with its three medians drawn, with the intersection points being $D, E, F,$ corresponding to $AB, BC,$ and $AC$ respectively. Thus, if we label point $A$ with a weight of $1$, $B$ must also have a weight of $1$ since $A$ and $B$ are equidistant to $D$. By the same process, we find $C$ must also have a weight of 1. Now, since $A$ and $B$ both have a weight of $1$, $D$ must have a weight of $2$ (as is true for $E$ and $F$). Thus, if we label the centroid $P$, we can deduce that $DP:PC$ is $1:2$ - the inverse ratio of their weights.


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

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