### 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 $a+b = c+d$.

### 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.