Mass point geometry

Mass point geometry, colloquially known as mass points, is a problem-solving technique in geometry 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 many students prefer to use mass points.

Though modern mass point geometry was developed in the 1960s by New York high school students,[3] the concept has been found to have been used as early as 1827 by August Ferdinand Möbius in his theory of homogeneous coordinates.

[4] The theory of mass points is defined according to the following definitions:[5] 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.

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.

intersect at

intersects

{\displaystyle {\tfrac {OB}{OE}}}

{\displaystyle {\tfrac {OD}{OA}}}

We may arbitrarily assign the mass of point

By ratios of lengths, the masses at

By summing masses, the masses at

, making the mass at

{\displaystyle {\tfrac {OB}{OE}}}

{\displaystyle {\tfrac {OD}{OA}}={\tfrac {3}{2}}}

{\displaystyle AB}

intersect at