Below is a diagram of a square, where $E, H, F,$ and $G$ are the midpoints of the square. I want to prove that the smaller square formed by the intersections of $EC, FD, BH,$ and $AG$ is a square.
So far, I see a lot of right angles but I can't seem to manipulate them. I tried to use coordinate geometry, like settings a point for $D$ and another point for $F.$ I also know the altitude to hypotenuse of a right triangle splits it into three similar triangles, but that isn't really useful, since we want to show that when the altitude is dropped, it will intersect the square at another midpoint. Could I have some help? Thanks in advance
Here's a more abstract approach. Consider the rotation $R$ that takes $A$ to $B$, $B$ to $C$, and so on. The square $ABCD$ is obviously invariant under this rotation.
In fact, the entire construction of the inner quadrilateral is also invariant under $R$: for example, $R$ takes $E$ to $F$, $F$ to $G$, and so on.
That means the inner quadrilateral itself is also invariant under $R$. So its four sides and four angles are all equal, which means it's a square.