Volume in Pyramids

[superchilangomex]

Why, in a regular triangular pyramid, does the altitude from the vertex to the base (height) of the pyramid), intersects the altitude (the altitude on the base that is an equilateral triangle) in a way that that segments gets cut in two parts that one is 2/3 of the total segment, and the other is 1/3 of the total segment?

 

[tkhunny]

I think you're question has absolutely nothing to do with pyramids.

Draw an equilateral triangle.
Draw the three altitudes (or the three angle bisectors).
Now answer the same question about 2/3 and 1/3 and the point of intersection of the altitudes.

Put the equilateral triangle on a set of coordiante axes and construct the equations of the diagonals and see where they intersect.

Vector methods are simpler, I think, but I'll leave that to you.