We assume vector addition and multiplication of a vector by a scalar (i.e., a real number) are well-understood.
Notation: Let I, J, K be (1, 0, 0), (0, 1, 0), (0, 0, 1).
The center of mass of a set of points (with equal masses at each point) is the mean, or average. Thus for two points, AB, the midpoint is (1/2)(A+B), for a triangle ABC, the center of mass (the centroid) is (1/3)(A + B + C). For a set of four points ABCD, the center of mass is (1/4)(A+B+C+D).
For an equilateral triangle, all the usual centers (centroid, circumcenter, incenter, orthocenter) are all the same point.
Given two points A and B and two masses a, b at those points, the center of mass is the weighted average:
M = (a/(a+b))A + (b/(a+b))B.
If a+b = 1, then M = aA + bB. In the case a+b = 1, ( if we set b = t, then a = 1 - t, so the formula for the center of mass is simpler:
P(t) = (1-t)A + t.B.
The formula P(t) = (1-t)A + t.B can be written in coordinates as, e.g., (x(t), y(t), z(t)). This is the (affine) parametrization of the line AB. The parameter t provides a coordinate system on the line AB.
When both masses are nonnegative, then P is on the segment AB. If one of the masses is negative, then P is on line AB but not on the segment.
For 3 points A, B, C, with masses a, b, c, the center of mass, or barycenter is
M = (1/(a+b+c))(aA+bB+cC).
Again this is simpler if a+b+c = 1, so the point M = aA+bB+cC. Finally, if b = s and c = t, then if a = 1 - s - t the sum = 1.
P(s,t) = (1 - s - t)A + sB + tC
The point P(s,t) = (1 - s - t)A + sB + tC ranges over the whole plane ABC as s and t take on all real values, with P inside triangle ABC if all 3 masses are nonnegative. P(s,t) can be written in coordinates as, e.g., (x(s,t), y(s,t), (s,t)).This is the (affine) parametrization of the plane ABC.
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