Incircle of a quadrilateral

One of the four circles below is the incircle of the quadrilateral: the largest circle that will fit within the four-sided shape. Construction:
  1. Extend the sides of the quadrilateral to form two triangles
  2. Draw the incircles of these two triangles (the centres are the intersections of the angle bisectors]
  3. At least one of these two circles will lie within the quadrilateral (prove this!).
  4. 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.