What happens to the perimeter of triangle ABC as the two smaller
circles change size and roll around inside the bigger circle?
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?
Show that for any triangle it is always possible to construct 3
touching circles with centres at the vertices. Is it possible to
construct touching circles centred at the vertices of any polygon?
A point $P$ is selected anywhere inside an equilateral triangle.
What can you say about the sum of the perpendicular distances from
$P$ to the sides of the triangle? Can you prove your
If you're stuck why not try
this proof sorting activity? Cut out the pieces and see if you can
rearrange them into a coherent proof.
If you have Java enabled, you can experment with the interactive
diagram below by clicking and dragging the red points