Begin by drawing a circle (center A and point B)
Create a point C on the circle
Draw a 2nd circle centered at B out to point A
Draw a 3rd circle centered at C out to point A
Create a point D at the intersection of these two circles
ABCD are the vertices of the rhombus
Finish by hiding the circles and drawing the line segments
Manipulate your rhombus and observe the behavior

Reset construction.
Part II
Begin by drawing a circle (center A and point B)
Create a point C on the circle
Draw line segments AB and AC
Construct a line through B parallel the AC
Construct a line through C parallel the AB
Create a point D at the intersection of these two lines
ABCD are the vertices of the rhombus
Finish by hiding the lines and drawing the line segments
Manipulate your rhombus and observe the behavior
Do you have any ideas on creating a different rhombus with the tools available.