What is the Set Intersection of Successive Midpoint Triangles in R^2?

[rakehsoran]

Hi
some one please help me with the following problem

Suppose that T_0 is the interior of a triangle in R^2 with vertices A,B,C. If T_1 is the interior of the trianlge whose vertices are midpoints of the sides of T_0, T_2 the intrior of the triangle whose vertices are midpoints of sides of T_1 and so on then what is in the set intersection(from j=0 to inf) T_j?

Thanks
Rakesh

 

[HallsofIvy]

It will be, of course, a single point. I think the centroid of the triangle.

 

[micromass]

What exactly does this have to do with functional analysis?

 

[mathwonk]

indeed it is the centroid. consider two medians of the triangle. they meet somewhere inside the triangle T0. in fact the same two medians are also medians of the triangle T1, so thy also meet inside that triangle, etc...thus the two medians meet at the unique point common to all these triangles. the same statement holds for any two medians. hence all three medians meet at the common point of all the triangles, which is therefore the centroid.