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 c ...
which applies the physical principle of the
center of mass
In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may ...
to geometry problems involving
triangle
A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), 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, an ...
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 Giovann ...
s. All problems that can be solved using mass point geometry can also be solved using either
similar triangle
In Euclidean geometry, two objects are similar if they have the same shape, or one has the same shape as the mirror image of the other. More precisely, one can be obtained from the other by uniformly scaling (enlarging or reducing), possibly wi ...
s,
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 his ...
in his theory of
homogeneous coordinates
In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometry. T ...
.
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
, also written as
, including a mass,
, and an ordinary point,
on a plane.
* Coincidence - We say that two points
and
coincide if and only if
and
.
* Addition - The sum of two mass points
and
has mass
and point
where
is the point on
such that
. In other words,
is the fulcrum point that perfectly balances the points
and
. An example of mass point addition is shown at right. Mass point addition is
closed
Closed may refer to:
Mathematics
* Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set
* Closed set, a set which contains all its limit points
* Closed interval, ...
,
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 o ...
, and
associative
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement f ...
.
* Scalar Multiplication - Given a mass point
and a positive real
scalar
Scalar may refer to:
*Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers
* Scalar (physics), a physical quantity that can be described by a single element of a number field such ...
, we define multiplication to be
. 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 A ...
, 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
In geometry, the angle bisector theorem is concerned with the relative lengths of the two segments that a triangle's side is divided into by a line that bisects the opposite angle. It equates their relative lengths to the relative lengths of th ...
.
* Stewart's theorem - When asked not for the ratios of lengths but for the actual lengths themselves,
Stewart's theorem
In geometry, Stewart's theorem yields a relation between the lengths of the sides and the length of a cevian in a triangle. Its name is in honour of the Scottish mathematician Matthew Stewart, who published the theorem in 1746.
Statement
Let ...
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
,
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.
* System One. For the first system, we will choose
as our central point, and we may therefore ignore segment
and points
,
, and
. We may arbitrarily assign the mass at
to be
, and by ratios of lengths the masses at
and
are
and
, respectively. By summing masses, we get the masses at
,
, and
to be 10, 9, and 13, respectively. Therefore,
and
.
* System Two. For the second system, we will choose
as our central point, and we may therefore ignore segment
and points
and
. As this system involves a transversal, we must use split masses on point
. We may arbitrarily assign the mass at
to be
, and by ratios of lengths, the mass at
is
and the mass at
is split
towards
and 2 towards
. By summing masses, we get the masses at
,
, and
to be 4, 6, and 10, respectively. Therefore,
and
.
* 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:
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 Giovann ...
*
Ceva's theorem
In Euclidean geometry, Ceva's theorem is a theorem about triangles. Given a triangle , let the lines be drawn from the vertices to a common point (not on one of the sides of ), to meet opposite sides at respectively. (The segments are kn ...
*
Menelaus's theorem
Menelaus's theorem, named for Menelaus of Alexandria, is a proposition about triangles in plane geometry. Suppose we have a triangle ''ABC'', and a transversal line that crosses ''BC'', ''AC'', and ''AB'' at points ''D'', ''E'', and ''F'' respe ...
*
Stewart's theorem
In geometry, Stewart's theorem yields a relation between the lengths of the sides and the length of a cevian in a triangle. Its name is in honour of the Scottish mathematician Matthew Stewart, who published the theorem in 1746.
Statement
Let ...
*
Angle bisector theorem
In geometry, the angle bisector theorem is concerned with the relative lengths of the two segments that a triangle's side is divided into by a line that bisects the opposite angle. It equates their relative lengths to the relative lengths of th ...
*
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 A ...
*
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 div ...
Notes
{{commons category
Geometric centers
Triangle geometry