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Mass point geometry, colloquially known as mass points, is a problem-solving technique in
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
which applies the physical principle of the center of mass to geometry problems involving
triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colline ...
s and intersecting
cevian In geometry, a cevian is a line that intersects both a triangle's vertex, and also the side that is opposite to that vertex. Medians and angle bisectors are special cases of cevians. The name "cevian" comes from the Italian mathematician Giovan ...
s. All problems that can be solved using mass point geometry can also be solved using either similar triangles, vectors, or area ratios, but many students prefer to use mass points. Though modern mass point geometry was developed in the 1960s by New York high school students, the concept has been found to have been used as early as 1827 by
August Ferdinand Möbius August Ferdinand Möbius (, ; ; 17 November 1790 – 26 September 1868) was a German mathematician and theoretical astronomer. Early life and education Möbius was born in Schulpforta, Electorate of Saxony, and was descended on hi ...
in his theory of homogeneous coordinates.


Definitions

The theory of mass points is defined according to the following definitions:H. S. M. Coxeter, ''Introduction to Geometry'', pp. 216-221, John Wiley & Sons, Inc. 1969 * Mass Point - A mass point is a pair (m, P), also written as mP, including a mass, m, and an ordinary point, P on a plane. * Coincidence - We say that two points mP and nQ coincide if and only if m = n and P = Q. * Addition - The sum of two mass points mP and nQ has mass m + n and point R where R is the point on PQ such that PR:RQ = n:m. In other words, R is the fulcrum point that perfectly balances the points P and Q. An example of mass point addition is shown at right. Mass point addition is closed,
commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
, and associative. * Scalar Multiplication - Given a mass point mP and a positive real scalar k, we define multiplication to be k(m, P) = (km, P). Mass point scalar multiplication is distributive over mass point addition.


Methods


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

* Routh's theorem - Many problems involving triangles with cevians will ask for areas, and mass points does not provide a method for calculating areas. However,
Routh's theorem In geometry, Routh's theorem determines the ratio of areas between a given triangle and a triangle formed by the pairwise intersections of three cevians. The theorem states that if in triangle ABC points D, E, and F lie on segments BC, CA, and ...
, which goes hand in hand with mass points, uses ratios of lengths to calculate the ratio of areas between a triangle and a triangle formed by three cevians. * Special cevians - When given cevians with special properties, like an
angle bisector In geometry, bisection is the division of something into two equal or congruent parts, usually by a line, which is then called a ''bisector''. The most often considered types of bisectors are the ''segment bisector'' (a line that passes through ...
or an
altitude Altitude or height (also sometimes known as depth) is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The exact definition and reference datum varies according to the context ...
, other theorems may be used alongside mass point geometry that determine length ratios. One very common theorem used likewise is the angle bisector theorem. * Stewart's theorem - When asked not for the ratios of lengths but for the actual lengths themselves, Stewart's theorem may be used to determine the length of the entire segment, and then mass points may be used to determine the ratios and therefore the necessary lengths of parts of segments. * Higher dimensions - The methods involved in mass point geometry are not limited to two dimensions; the same methods may be used in problems involving tetrahedra, or even higher-dimensional shapes, though it is rare that a problem involving four or more dimensions will require use of mass points.


Examples


Problem One

Problem. In triangle ABC, E is on AC so that CE = 3AE and F is on AB so that BF = 3AF. If BE and CF intersect at O and line AO intersects BC at D, compute \tfrac and \tfrac. Solution. We may arbitrarily assign the mass of point A to be 3. By ratios of lengths, the masses at B and C must both be 1. By summing masses, the masses at E and F are both 4. Furthermore, the mass at O is 4 + 1 = 5, making the mass at D have to be 5 - 3 = 2 Therefore \tfrac = 4 and \tfrac = \tfrac. See diagram at right.


Problem Two

Problem. In triangle ABC, D, E, and F are on BC, CA, and AB, respectively, so that AE = AF = CD = 2, BD = CE = 3, and BF = 5. If DE and CF intersect at O, compute \tfrac and \tfrac. Solution. As this problem involves a transversal, we must use split masses on point C. We may arbitrarily assign the mass of point A to be 15. By ratios of lengths, the mass at B must be 6 and the mass at C is split 10 towards A and 9 towards B. By summing masses, we get the masses at D, E, and F to be 15, 25, and 21, respectively. Therefore \tfrac = \tfrac = \tfrac and \tfrac = \tfrac = \tfrac.


Problem Three

Problem. In triangle ABC, points D and E are on sides BC and CA, respectively, and points F and G are on side AB with G between F and B. BE intersects CF at point O_1 and BE intersects DG at point O_2. If FG = 1, AE = AF = DB = DC = 2, and BG = CE = 3, compute \tfrac. Solution. This problem involves two central intersection points, O_1 and O_2, so we must use multiple systems. * System One. For the first system, we will choose O_1 as our central point, and we may therefore ignore segment DG and points D, G, and O_2. We may arbitrarily assign the mass at A to be 6, and by ratios of lengths the masses at B and C are 3 and 4, respectively. By summing masses, we get the masses at E, F, and O_1 to be 10, 9, and 13, respectively. Therefore, \tfrac = \tfrac and \tfrac = \tfrac. * System Two. For the second system, we will choose O_2 as our central point, and we may therefore ignore segment CF and points F and O_1. As this system involves a transversal, we must use split masses on point B. We may arbitrarily assign the mass at A to be 3, and by ratios of lengths, the mass at C is 2 and the mass at B is split 3 towards A and 2 towards C. By summing masses, we get the masses at D, G, and O_2 to be 4, 6, and 10, respectively. Therefore, \tfrac = \tfrac = 1 and \tfrac = \tfrac. * Original System. We now know all the ratios necessary to put together the ratio we are asked for. The final answer may be found as follows: \tfrac = \tfrac = 1 - \tfrac - \tfrac = 1 - \tfrac - \tfrac = \tfrac.


See also

*
Cevian In geometry, a cevian is a line that intersects both a triangle's vertex, and also the side that is opposite to that vertex. Medians and angle bisectors are special cases of cevians. The name "cevian" comes from the Italian mathematician Giovan ...
* Ceva's theorem * Menelaus's theorem * Stewart's theorem * Angle bisector theorem *
Routh's theorem In geometry, Routh's theorem determines the ratio of areas between a given triangle and a triangle formed by the pairwise intersections of three cevians. The theorem states that if in triangle ABC points D, E, and F lie on segments BC, CA, and ...
*
Barycentric coordinates In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related ...
*
Lever A lever is a simple machine consisting of a beam or rigid rod pivoted at a fixed hinge, or '' fulcrum''. A lever is a rigid body capable of rotating on a point on itself. On the basis of the locations of fulcrum, load and effort, the lever is d ...


Notes

{{commons category Geometric centers Triangle geometry