I've done constructing the oval, but now I'm having trouble drawing a square on it. What should I do about this? (The three points given inside the oval are two focuses and the midpoint of the focuses.)
I've tried drawing the meet of the focus-point line and oval(like this...)
But this has no reason, and I think it doesn't work...
Another thing I've thought about is drawing a square first and minimize it to the oval, but I don't have any idea to do like that.
You will find plainly the 4 vertices of the square by intersecting the ellipse with the 2 angular bissectors of the symmetry axes of the ellipse (in other terms lines with equations $y=x$ and $y=-x$ when $x$ and $y$ coordinate axes are the symmetry axes), all constructions involved (including the construction of the $y$ axis) being evidently possible by ruler and compass.
In this answer it is proven that no other inscribed square exists.