# Functional Analysis doubt

1. Oct 31, 2012

### rakehsoran

Hi

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

2. Oct 31, 2012

### HallsofIvy

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

3. Oct 31, 2012

### micromass

What exactly does this have to do with functional analysis???

4. Nov 5, 2012

### 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.