Difference between revisions of "Mass points"
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| − | Throughout this solution, let <math>W_{X}</math> denote the weight at point <math>X</math>. Since <math>CE | + | Throughout this solution, let <math>W_{X}</math> denote the weight at point <math>X</math>. Since <math>\frac{CE}{EB} = \frac{8}{3}</math>, let <math>W_{B} = 8</math>, which makes <math>W_{C} = 3</math>. Now, look at <math>AB</math>. Since <math>W_{B} \cdot BD = W_{A} \cdot AD</math> (this is a general property commonly used in many mass points problems, in fact it is the same property we used above to determine <math>W_{B}</math>), we have <math>W_{A} = W_{B} \cdot \frac{BD}{AD} = 8 \cdot \frac{5}{3} = \frac{40}{3}</math>. Then, <math>W_{F} = W_{A} + W_{C} = \frac{40}{3} + 3 = \frac{49}{3}</math> (another property of mass points). Finally, we have <math>FG:GB = W_{B}:W_{F} = 8:\frac{49}{3} = \boxed{24:49}</math>. |
~advanture | ~advanture | ||
Revision as of 23:00, 12 March 2021
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 was invented by Franz Mobius in 1827 along with his theory of homogeneous coordinates. The technique did not catch on until the 1960s when New York high school students made it popular. The technique greatly simplifies certain problems. 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 inversely 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).
The way to systematically assign weights to the points involves first choosing a point for the entire figure to balance around. From there, WLOG a first weight can be assigned. From the first weight, others can be derived using a few simple rules. Any line passing this central point will balance the figure. If two points balance, the product of the mass and distance from a line of balance of one point will equal the product of the mass and distance from the same line of balance of the other point. If two points are balanced, the point on the balancing line used to balance them has a mass of the sum of the masses of the two points. That is, if two points (
and 
) have masses 
and 
, respectively, a third point between and which divides 
into the ratio 
will have mass 
.
Examples
Example 1
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 from 
. 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.
Example 2
has point on , point on 
, and point on . 
, 
, and 
intersect at point 
. The ratio 
is 
and the ratio 
is 
. Find the ratio of 
Solution
Throughout this solution, let 
denote the weight at point 
. Since 
, let 
, which makes 
. Now, look at . Since 
(this is a general property commonly used in many mass points problems, in fact it is the same property we used above to determine 
), we have 
. Then, 
(another property of mass points). Finally, we have 
.
~advanture
Problems
2019 AMC 8 Problems/Problem 24
2016 AMC 10A Problems/Problem 19
2013 AMC 10B Problems/Problem 16
2004 AMC 10B Problems/Problem 20
2016 AMC 12A Problems/Problem 12
2009 AIME I Problems/Problem 5
2009 AIME I Problems/Problem 4
2001 AIME I Problems/Problem 7
2011 AIME II Problems/Problem 4
Video Lecture
The Central NC Math Group recently released a lecture on Mass Points and Barycentric Coordinates, which you can view at https://www.youtube.com/watch?v=KQim7-wrwL0.
