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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.
This printable worksheet may be useful: Circles in Quadrilaterals.
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.
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.
If the side lengths of a tangential quadrilateral are $a$, $b$, $c$ and $d$, with $a$ opposite $c$ and $b$ opposite $d$, show that $a+b = c+d$.