Dear forum members, I am currently doing a problem related to hexagons.

We have a regular hexagon. By joining the midpoints of the sides we get another hexagon inside of the original, and by joining the midpoints of the sides of the smaller hexagon we get another one and so forth. I am told to calculate the first hexagon, whose area is less than one millionth of the original. I know how to do this, but I have trouble in formulating the equation.

The apothem of the original hexagon would be given by $\displaystyle \frac{\sqrt{3}}{2}*a$, right?

Then comes something I can't get through my head. the book answer tells me that when I connect the two radii of the circumcircle of the original hexagon, I get an equilateral triangle, yet the book does not prove it. Is this really true?

And then the book tells me that the apothem of the orinal hexagon, equals the lenght of the side of the hexagon drawn inside the original hexagon. Can someone verify this or prove it, please. My thick head refuses to accept it.

Thank you!