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# Circles in Quadrilaterals

#### For each of the following types of quadrilaterals, decide whether it is always, sometimes or never possible to construct a circle inside which just touches all four sides:

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Age 14 to 16

Challenge Level

*Circles in Quadrilaterals printable sheet*

You may have come across the idea of a cyclic quadrilateral, where it is possible to draw a circle around the quadrilateral so that the circumference passes through all four vertices of the shape.

A **tangential quadrilateral** is one where it is possible to draw a circle inside it so that the circumference just touches all four sides of the shape.

Here are some examples of tangential quadrilaterals:

- Square
- Rectangle
- Rhombus
- Parallelogram
- Kite
- Trapezium

If you decide always or never, you need to justify your decision with a convincing argument.

If you decide sometimes you need to be precise about when it is possible and when it is not possible, and why.

Four rods are hinged at their ends to form a convex quadrilateral. Investigate the different shapes that the quadrilateral can take. Be patient this problem may be slow to load.

A kite shaped lawn consists of an equilateral triangle ABC of side 130 feet and an isosceles triangle BCD in which BD and CD are of length 169 feet. A gardener has a motor mower which cuts strips of grass exactly one foot wide and wishes to cut the entire lawn in parallel strips. What is the minimum number of strips the gardener must mow?

What can you say about the lengths of the sides of a quadrilateral whose vertices are on a unit circle?