Mass point geometry - AbsoluteAstronomy.com

Mass point geometry, colloquially known as mass points, is a geometry problem-solving technique which applies the physical principle of the center of mass

Center of mass

In physics, the center of mass or barycenter of a system is the average location of all of its mass. In the case of a rigid body, the position of the center of mass is fixed in relation to the body...

to geometry problems involving triangles and intersecting cevian

Cevian

In geometry, a cevian is any line segment in a triangle with one endpoint on a vertex of the triangle and the other endpoint on the opposite side. Medians, altitudes, and angle bisectors are special cases of cevians...

s. All problems that can be solved using mass point geometry can also be solved using either similar triangles, vectors, or area ratios, but mass point geometry is far quicker than those methods 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, the concept has been found to have been used as early as 1827 by August Ferdinand Möbius

August Ferdinand Möbius

August Ferdinand Möbius was a German mathematician and theoretical astronomer.He is best known for his discovery of the Möbius strip, a non-orientable two-dimensional surface with only one side when embedded in three-dimensional Euclidean space. It was independently discovered by Johann Benedict...

in his theory of homogenous coordinates.

Definitions

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

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
Closure (mathematics)
In mathematics, a set is said to be closed under some operation if performance of that operation on members of the set always produces a unique member of the same set. For example, the real numbers are closed under subtraction, but the natural numbers are not: 3 and 8 are both natural numbers, but...

, idempotent, commutative, and associative.
Scalar Multiplication - Given a mass point and a positive real scalar , we define multiplication to be . Mass point scalar multiplication is distributive over mass point addition.

Concurrent cevians

The method of using mass point geometry to solve problems with concurrent cevians is quite simple, and does not require much knowledge of the theory behind it. To begin with, a single point is assigned an arbitrary mass, usually one that allows the other masses in the problem to be integral. The masses at the other points are calculated so that the feet of cevians are the sum of the two mass point vertices they are between. For each cevian, the point of concurrency is the sum of the mass point vertex and foot of that cevian. 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

Transversal (geometry)

In geometry, a transversal is a line that passes through two or more other lines in the same plane at different points. When the lines are parallel, as is often the case, a transversal produces several congruent and several supplementary angles...

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
Routh's theorem
In geometry, Routh's theorem determines the ratio of areas between a given triangle and a triangle formed by the intersection of three cevians...

, 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 or an altitude
Altitude
Altitude or height is defined based on the context in which it is used . As a general definition, altitude is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The reference datum also often 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
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 the other two sides of the triangle.Consider a triangle...

.
Stewart's theorem - When asked not for the ratios of lengths but for the actual lengths themselves, Stewart's theorem
Stewart's theorem
In geometry, Stewart's theorem yields a relation between a lengths of the sides of the triangle and the length of a cevian of the triangle. Its name is in honor of the Scottish mathematician Matthew Stewart who published the theorem in 1746.- 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

is on

so that

and

is on

so that

. If

and

intersect at

and line

intersects

, 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

, 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 .