Center mass of triangle in 3-D space

[cue928]

Homework Statement

So I have a triangle with points: A (4,2,0), B (3,3,0), and C (1,1,3). We are to find the point at which the three medians intersect in i,j,k format. I've found the midpoints of each side but I don't know where to go from there.

 

[micromass]

Construct the equation of the line joining the midpoint and the opposite vertex. Do this for all vertices.

 

[dynamicsolo]

You can save a good bit of this work if you are allowed to use the theorem which states that the medians of a triangle intersect at a point with divides each median in the ratio of 2:1 . Then you can choose any one median and find the point (by proportions) which is one-third of the way from the base to the opposite vertex. (Any other choice of a median to work with should give exactly the same result.)

I am presuming in this that the problem is not asking you to show that all three medians meet at the point with this property.

 

[HallsofIvy]

The simplest way to do this: the coordinates of the centroid of a triangle (not, strictly speaking, the "center of mass" because a geometric figure does not have 'mass') is the mean of the coordinates of the three vertices. That is, if the vertices of the triangle are at $(x_1, y_1, z_1)$, [itex](x_2, y_2, z_2)[itex], $(x_3, y_3, z_3)$, then the centroid is at
$$\left(\frac{x_1+ x_2+ x_3}{3}, \frac{y_1+y_2+y_3}{3}, \frac{z_1+z_2+z_3}{3}\right)$$

If you don't have that theorem, use that as a check.
(The average of the coordinates works for the two dimensional triangle or three dimensional tetrahedron but not for other figures.)