Start with square W X Y Z. Put a dot at the middle of each side and name them A (middle of WX), B (middle of XY), C (middle of YZ), and D (middle of ZW). We will show A-B-C-D is also a square.
- All four sides are the same length.
Imagine the big square has side length 2 (this keeps numbers small). Then each half of a side (from a corner to a midpoint) is length 1. Look at side AB: it goes from the middle of the bottom side to the middle of the right side. That short segment AB makes a little right triangle whose two legs are each 1. The slanted side AB (the hypotenuse) has the same length as every other slanted side made the same way, because every side of the big square is the same. So AB = BC = CD = DA. - Each corner of A-B-C-D is a right angle (90°).
AB slants up and to the right. BC slants up and to the left. When one side slants up-right and the next side slants up-left, they meet at a corner that is a right angle (just like two sides of a plus-sign "+"). The same happens at every corner because of the square's symmetry. So every corner of A-B-C-D is 90°. - Equal sides + right angles = square.
Because the four sides are equal in length and each corner is a right angle, the shape A-B-C-D must be a square. In fact, it's the original square turned a bit (rotated 45°) and sitting inside the big square.
So, whenever you join the midpoints of the sides of a square, the new four-sided shape is also a square.