I have several questions to verify and my fill the gaps in my knowledge on centers of mass in pyramids. I would be also very grateful if you could verify the informations I already have.
So, I know that the center of mass of tetrahedron is called a centroid. According to Wikipedia: "A line segment joining a vertex of a tetrahedron with the centroid of the opposite face is called a median and a line segment joining the midpoints of two opposite edges is called a bimedian of the tetrahedron. Hence there are four medians and three bimedians in a tetrahedron. These seven line segments are all concurrent at a point called the centroid of the tetrahedron. In addition the four medians are divided in a 3:1 ratio".
Now, I would like to expand that properties on other pyramids. Those are my questions.
1) Does the center of the mass of tetrahedron lay in the orthocenter?
1.1) Is there any pyramid other than the tetrahedron that has center of the mass and orthocenter in the same place (all regular pyramids? all regular triangular pyramids?)?
1.2) Is there any pyramid other than the tetrahedron that has a orthocenter (all regular pyramids? all regular triangular pyramids?)?
2) If I draw a line between the vortex and the center of mass of the base in ANY triangular pyramid will it be located in the crossing of all 4 medians and 3 bimedians? Will the piont of crossing devide medians in 3:1 ratio? I mean to apply this rule to the triangular pyramids in which the projection of the vertex on the base is not in the center of mass of the base. What about all regular triangular pyramids?
2.1) Does it work for every pyramid with odd number of edges in the base?
2.2) What about the ratio of bimedians? Is there any fixed ration of their division?
3) In the case of pyramids with an even number of edges in the base, according to the theory, there is just one median (joining a vertex of a tetrahedron with the centroid of the base). Therefore, it can not be crossed by any other. There are however 2 tipes of possible "bimedians" corssing the height of pyramid, each one of them: (1) joining the midpoint of the base edge with the midpoint of height in side face ; (2) joining the midpoint of the side edge with the vertex in the base.
Does one tipe of those bimedians devide the existing median in any fixed ratio? Does it work in ANY pyramid with an even number of edges in the base (regular ones? those with projection of the vertex on the base out of the center of mass of the base?)?
I would appreciate it if you could help me. I know that there are a lot of questions. You don't need to answer them all at once. Every answer would be helpful.