### Some(?) of the Parts

A circle touches the lines OA, OB and AB where OA and OB are perpendicular. Show that the diameter of the circle is equal to the perimeter of the triangle

### Rotating Triangle

What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle?

### Pericut

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:

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.