Mass point geometry

Mass point geometry, colloquially known as mass points, is a geometry problem-solving technique which applies the physical principle of the center of mass to geometry problems involving triangles and intersecting cevians.[1] All problems that can be solved using mass point geometry can also be solved using either similar triangles, vectors, or area ratios,[2] but mass point geometry is far quicker than those methods[3] and thus is used more often on math competitions in which time is an important factor. Though modern mass point geometry was developed in the 1960s by New York high school students,[4] the concept has been found to have been used as early as 1827 by August Ferdinand Möbius in his theory of homogeneous coordinates.[5]


Example of mass point addition

The theory of mass points is rigorously defined according to the following definitions:[6]


Concurrent cevians

First, a point is assigned with a mass (often a whole number, but it depends on the problem) in the way that other masses are also whole numbers. The principle of calculation is that the foot of a cevian is the addition (defined above) of the two vertices (they are the endpoints of the side where the foot lie). For each cevian, the point of concurrency is the sum of the vertex and the foot. Each length ratio may then be calculated from the masses at the points. See Problem One for an example.

Splitting masses

Splitting masses is the slightly more complicated method necessary when a problem contains transversals in addition to cevians. Any vertex that is on both sides the transversal crosses will have a split mass. A point with a split mass may be treated as a normal mass point, except that it has three masses: one used for each of the two sides it is on, and one that is the sum of the other two split masses and is used for any cevians it may have. See Problem Two for an example.

Other methods


Diagram for solution to Problem One
Diagram for solution to Problem Two
Diagram for Problem Three
Diagram for Problem Three, System One
Diagram for Problem Three, System Two

Problem One

Problem. In triangle , is on so that and is on so that . If and intersect at and line intersects at , compute and .

Solution. We may arbitrarily assign the mass of point to be . By ratios of lengths, the masses at and must both be . By summing masses, the masses at and are both . Furthermore, the mass at is , making the mass at have to be Therefore and . See diagram at right.

Problem Two

Problem. In triangle , , , and are on , , and , respectively, so that , , and . If and intersect at , compute and .

Solution. As this problem involves a transversal, we must use split masses on point . We may arbitrarily assign the mass of point to be . By ratios of lengths, the mass at must be and the mass at is split towards and towards . By summing masses, we get the masses at , , and to be , , and , respectively. Therefore and .

Problem Three

Problem. In triangle , points and are on sides and , respectively, and points and are on side with between and . intersects at point and intersects at point . If , , and , compute .

Solution. This problem involves two central intersection points, and , so we must use multiple systems.

See also


  1. Rhoad, R., Milauskas, G., and Whipple, R. Geometry for Enjoyment and Challenge. McDougal, Littell & Company, 1991.
  4. Rhoad, R., Milauskas, G., and Whipple, R. Geometry for Enjoyment and Challenge. McDougal, Littell & Company, 1991
  5. D. Pedoe Notes on the History of Geometrical Ideas I: Homogeneous Coordinates. Math Magazine (1975), 215-217.
  6. H. S. M. Coxeter, Introduction to Geometry, pp. 216-221, John Wiley & Sons, Inc. 1969
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