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# Strange Rectangle

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Age 16 to 18

Challenge Level

Show that, for any values of $x$ and $y$ where $x> y$,

- the angle $SPQ$ is always the same and find this angle
- the angle $SRQ$ is always the same and find this angle
- the quadrilateral $SPQR$ is cyclic
- the angle $PQR$ is always the same and find this angle
- the angle $PSR$ is always the same and find this angle.

A triangle PQR, right angled at P, slides on a horizontal floor with Q and R in contact with perpendicular walls. What is the locus of P?

Four rods are hinged at their ends to form a quadrilateral. How can you maximise its area?

Find the exact values of some trig. ratios from this rectangle in which a cyclic quadrilateral cuts off four right angled triangles.