Intersection of nested triangles is non-empty Let $T_0$ be the interior of a triangle in $\mathbb{R}^2$ with vertices $A,B,C.$ Let $T_1$ be the interior of the triangle whose vertices are the midpoints of the sides if $T_0$ and $T_2$ be the interior of the triangle whose vertices are the midpoints of the sides if $T_1$ and so on. Then find $$\bigcap_{i=0}^\infty T_i$$
My guess is that the intersection will contain only the centroid. Since the diameter of the triangle goes to zero so the intersection will contain only the centroid. But I am unable to give a concrete proof. 

What you've written basically gets you there already. It's easy to see that the centroid $x$ lies in $\bigcap T_i$ since it clearly lies in each $T_i$. Then as you note $\mbox{diam}(T_i) \to 0$ as $i \to \infty$. If there are two points $x,y \in \bigcap T_i$ then $x,y \in T_j$ for each $j$ so $\mbox{diam}(T_j) \geq |x-y| > 0$ for each $j$, giving a contradiction so $\bigcap T_i = \{x\}$
