Quadrilateral Formed by the Angle Bisectors of a Parallelogram

Begin with any parallelogram ABCD.

Construct the angle bisectors of each angle of ABCD. These four rays intersect in a way that forms a second quadrilateral which we will label EFGH.

Begin exploring the properties of EFGH and its relationship to parallelogram ABCD by varying the properties of the given ABCD.

 

General Observations:

1. The four angle bisectors always intersect to form a convex quadrilateral EFGH as long as ABCD is a generic parallelogram (i.e. no right angles, adjacent sides not congruent).

2. EFGH is always a rectangle.

3. EFGH sometimes lies entirely inside of parallelogram ABCD as seen above; at other times EFGH overlaps ABCD and extends outside as below; however, ABCD never lies entirely inside of EFGH. This is because the angle bisector ray must lie between the rays containing adjacent sides of the parallelogram.

4. The diagonals of EFGH are parallel to the sides of ABCD. The diagonals of EFGH bisect pairs of opposite sides of ABCD.

  1. There is no clear relationship between the ratio of the lengths of adjacent sides of ABCD and EFGH.

What happens when ABCD is a special paralelogram?

  1. Rhombus

    2. EFGH disappears because opposite angles share angle bisectors.

  2. Rectangle

EFGH is a square with diamond orientation.