The length and the properties of a bisector of a parallelogram

Details
1.  Properties of a bisector of a parallelogram  
By definition, the bisector divides the angle in halves  :
The bisector cuts out an isosceles triangle (in this case, the triangles ABF and DKC)  :
The bisectors of the supplementary angles intersect at a right angle  :
The bisectors of the opposite angles are equal and parallel  :

 

bisector parallelogram

    A16 brown F16 violet     -  bisector of an acute angle

   D16 brown K16 violet   -  bisector of an obtuse angle

   a black , b black    -  sides of a parallelogram

  alpha red  -  acute angle

  beta2 blue  -  obtuse angle

 

Since ABF and DKC are isosceles triangles, the following identities hold  :

bisector parallelogram F1

bisector parallelogram F2

 

 

2.  The length of a bisector of a parallelogram

length bisector parallelogram

 

    a black , b black      -  sides of a parallelogram

     alpha red  -  acute angle

     beta2 blue  -  obtuse angle

   L  - bisector

 

Find length of bisector of a parallelogram if given side and angle ( L  ) :

length bisector parallelogram F1

length bisector parallelogram F12

length bisector parallelogram F2