A standard result in geometry is that the medians of a triangle meet at a point and cut each other
in the ratio of 2:1
Suppose we subdivide each leg of the triangle in two parts so that the ratio of the two parts is not 1:1 as in the case of medians, but rather n:1. Then we draw lines from the vertices to these points - let's call these lines n-dians.
The three n-dians no longer intersect at a point - rather they define a triangle. What is the relationship between this triangle and the original triangle?
What can you say about the lengths of the segments determined by the intersections of the n-dians?
Can you prove {some, all} of your assertions?