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## 'Darts and Kites' printed from http://nrich.maths.org/

The diagram shows a rhombus $PQRS$ with an internal point $O$
such that $OQ = OR = OS = 1$ unit. Penrose used this rhombus, split
into two quadrilaterals, a dart and a kite, to make his famous
tiling which fills the plane but, unlike a tessellation, does not
repeat itself by translation or rotation.

Find all the angles in the diagram, show that $POR$ is a
straight line and show that triangles $PRS$ and $QRO$ are similar.
Hence prove that the length of the side of the rhombus is equal to
the Golden Ratio $(1+ \sqrt{5})/2$.