Consider an equilateral triangle with the equal sides each h

Consider an equilateral triangle with the equal sides each having length 2b (see below). Remove a smaller equilateral triangle (upside down) and find the area of the original triangle and the removed triangle (in terms of b). The removal leaves 3 smaller right-side-up triangles. Remove 3 upside down triangles as before and compute the area of the total removed. Continue removing the correct number of triangles and compute the area removed at the nth step. Write the sum of all areas removed as an infinite series. Find the sum of this infinite series, and thus the total area removed from the original triangle. Is every point on the original triangle removed? Why or why not? Explain.

Solution

Some Hints: It\'s at least clear that (1/4)th of the area is removed at first, and then 3*(1/4)th of the area of the area removed is removed on the second step. Also, it is clear that the vertices of the triangle are never removed. And not just the vertices, but also the midpoint of each edge length will never be removed either.