If the respective incentres, P, Q, R and S of triangles ABC, BCD, CDA and DAB of a cyclic quadrilateral ABCD are constructed, then PQRS is a rectangle.
Cyclic Quadrilateral Incentre-Rectangle
Can you prove the result? Note that when ABCD becomes crossed, PQRS also becomes a 'crossed rectangle', but still has two axes of symmetry through its two pairs of opposite sides (though no longer all angles are equal as in the traditional schoolbook definition of rectangles since a crossed quadriletaral has 2 reflex and 2 non-reflex angles).
Hint: Trying proving the result using the two results respectively at LTM Cover Problem and Triangle Circumcircle-Incentre Result
A proof of the result for the convex case (and adaptible for the crossed case) is given on p. 189-191 of Some Adventures in Euclidean Geometry, which is available for purchase as downloadable PDF or printed book at More Info
This result also appears in Harold Jacobs' Geometry and in HSM Coxeter's Introduction to geometry. Peter Ash also revisited this result recently at his blog Math Ed Blog, and where a link to his proof is given.
Michael de Villiers, Updated 23 July 2012.