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Two semicircle sit on the diameter of a semicircle centre O of twice their radius. Lines through O divide the perimeter into two parts. What can you say about the lengths of these two parts?

Circles in Quadrilaterals

Stage: 4 Challenge Level: Challenge Level:1

Why do this problem?

This problem provides an opportunity for rich discussion of properties of quadrilaterals and circles, and leads to geometrical reasoning in searching for proofs and counter-examples.

Possible approach

Show the three examples of tangential quadrilaterals and allow the learners to identify what they have in common. Share the definition of a tangential quadrilateral as one where a circle can be constructed inside to just touch all four sides.

Allow plenty of time to experiment - this could be through sketches on mini whiteboards or rough paper at first, leading to more accurate construction using ruler and compasses or using dynamic geometry. Encourage pairs to share their conjectures about in which quadrilaterals it will be always, sometimes or never possible to inscribe a circle.

Bring the class together to share their thoughts. Expect some disagreement, particularly for those shapes which are only sometimes possible. Give learners time to work in small groups to produce posters showing some or all of the quadrilaterals with reasoned arguments to explain their conclusions about whether they are tangential quadrilaterals or not, and then ask groups to present their conclusions to each other, encouraging healthy debate where disagreement still exists.

Key questions

When is it possible to draw a circle inside a kite? a trapezium?
Are there any quadrilaterals where it is never possible to inscribe a circle?

Possible extension

If the side lengths of a tangential quadrilateral are $a$, $b$, $c$ and $d$, with $a$ opposite $c$ and $b$ opposite $d$, show that $a+b = c+d$.

Bicentric Quadrilaterals builds on the ideas discussed in this problem and links with work on cyclic quadrilaterals.

Possible support

Create lots of diagrams to build up ideas of what is and isn't possible. There is a diagram in the Hint showing a semicircle constructed in a triangle; considering this may help for those quadrilaterals which can be cut along a line of symmetry into two triangles.