grid kites and rhombuses

Pierre van Hiele advocates constructing the special quadrilaterals by means of reflections and rotations
that way the properties are easily deduced (e.g. "because of the mirror line") from the constructions

this has been explored more fully by Michael Villiers in South Africa, using an interactive geometry package

a powerpoint goes through the constructions (but it's probably better demonstrated with an interactive package)

rhombus area

using areas of triangles to work out areas of rhombuses
possibly generalising