I need to prove that the medians of a tetrahedron are concurrent, and to find the ratio at which they intersect each other. I cannot use coordinate geometry, and I must use barycentric arguments in my solution.
Thus far, I have the following: Consider tetrahedron ABCD (with triangle ABC being the "bottom" triangle) with masses of 1 at each of the four vertices Consider face ABC, which has medians AD, BE, and CF, which intersects at the centroid of ABC, M. Each median is divided, by M, into segments of ratio 2:1. Further, M has a mass of 3.
Construct DM, the median of the tetrahedron from vertex D to centroid M of ABC> Also, construct CN, the median of the tetrahedron from vertex C to centroid N on triangle ABD.
I'm confident that CN and DM intesect, but I'm not sure why (I know they're on the same plane but I'm having trouble "seeing" it). In addition, I'm not certain how to go about showing that the other two medians are concurrent with these two.
I'm also pretty sure that the ratio that I'm looking for is 3:1, but I'm not sure how to go about showing it; the center of mass of the tetrahedron would have a mass of 4 since there are masses of 1 at each vertex, but I'm not sure how to go from there either.