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Circles in Quadrilaterals

Stage: 4 Challenge Level: Challenge Level:1

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:


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.

  • 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.

Send us your explanations and justifications, and anything else of interest you discover about these special quadrilaterals!