## Cyclic Quadrilateral Incentre-Rectangle

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.

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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.

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Michael de Villiers, Updated 23 July 2012.