1) Equal sides

Under a reflection around the axis of symmetry x, side AB maps to side , and side BC maps to side . Hence, side AB = side and side BC = side . Similarly, by reflection around the axis of symmetry it follows that = CB and = DC. By the property it therefore follows that AB = BC = CD = DA; i.e. all sides are equal.

2) Perpendicular diagonals

A reflection of triangle ABI around the axis of symmetry x maps it to triangle . Since diagonal BD, which lies on y, is a straight line, it follows that adjacent angles AIB and are 90 degrees. Similarly, from a around line y it follows that angle AIB = 90 = angle .

3) Bisected opposite angles

A reflection of angle BAC around the axis of symmetry x maps it to angle . Hence, diagonal AC angle BAD. In the same, it follows that the other angles are bisected by the diagonals.

4) Circumscribed

The line of symmetry x is the bisector of both angles BAD and BCD, and so is line of angles ABC and CDA. Since, an angle bisector is the path of all points equi-distant from the forming the angle it bisects, it follows that since all 4 angle bisectors are concurrent at I, that I is from sides AB, BC, CD and DA. Hence, a circle can be drawn with I as and one of the four equal distances to the four sides as the radius.