Quadrilateral that Can Inscribe a Circle

Move the green points to change the shape of the quadrilateral. Move the red point to change the size of the circle. Move the blue points to change the path restriction of the green points.
(1) Aside from irregular quadrilaterals, what type of quadrilaterals can do the following: a. always inscribe a circle b. sometimes inscribe a circle c. cannot inscribe a circle? (2). Justify your answers in (1).