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 points.

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:

circle in square

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.

Send us your explanations and justifications, and anything else of interest you discover about these special quadrilaterals!

smartphone. Tangents. Quadrilaterals. Mathematical reasoning & proof. Pythagoras' theorem. Visualising. Investigations. Circles. Circle theorems. Radius (radii) & diameters.