Circles in quadrilaterals
Explore when it is possible to construct a circle which just touches all four sides of a quadrilateral.
Problem
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:
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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
Getting Started
Here I have drawn a semicircle which just touches a triangle.
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Student Solutions
Preveina from Crest Girls' Academy sent us some pictures to support her reasoning about some of the shapes in this problem:
A circle can be always fitted in a square touching all 4 sides since the sides of a square are all equal. This makes the circle touch each side of the square evenly.
A circle can never be fitted in to a rectangle touching all 4 sides because a rectangle has 2 long sides and 2 short sides. When you're trying to draw a circle that touches all 4 sides in a rectangle it'll turn out to be an oval, since there are 2 long sides.
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Teachers' Resources
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
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. This is called an inscribed circle.
Key questions
When is it possible to draw a circle inside a kite? a trapezium?
Possible support
Create lots of diagrams to build up ideas of what is and isn't possible. There is a diagram in Getting Started showing a semicircle constructed in a triangle; considering this may be helpful for those quadrilaterals which can be cut along a line of symmetry into two triangles.
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 $a+c = b+d$.