### 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

### 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?

### Polycircles

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?

# Rotating Triangle

##### Stage: 3 and 4 Challenge Level:

Why do this problem?
The problem gives an opportunity for learners to make a conjecture and then to test whether their conjecture holds, and totry to prove it.

Possible approach
Ask learners to write down the lengths in the diagram in terms of the radii of the three circles. Let them introduce their own notation.

Key question
At a point of contact between two circles, the circles have a common tangent. What can you say about the lines from the centres of the two circles to the point of contact?

Possible extension
Download a free copy of the program Geogebra and let learners use it to experiment and to draw their own dynamic mathematics diagrams including the diagram from this question.