One of the four circles below is the incircle of the quadrilateral: the largest circle that will fit within the four-sided shape.
Construction:

Extend the sides of the quadrilateral to form two triangles

Draw the incircles of these two triangles (the centres are the intersections of the angle bisectors]

At least one of these two circles will lie within the quadrilateral (prove this!).

If there were a larger circle in the quadrilateral, then this circle would also be contained in both triangles, but be larger than the incircle of one. This gives a contradiction.

What is the connection with a cyclic quadrilateral?

Is there any significance to the other two circles?