### Just Touching

Three semi-circles have a common diameter, each touches the other two and two lie inside the biggest one. What is the radius of the circle that touches all three semi-circles?

Four rods are hinged at their ends to form a convex quadrilateral. Investigate the different shapes that the quadrilateral can take. Be patient this problem may be slow to load.

### Biggest Bendy

Four rods are hinged at their ends to form a quadrilateral with fixed side lengths. Show that the quadrilateral has a maximum area when it is cyclic.

# Calculating with Cosines

##### Stage: 4 and 5 Challenge Level:

Try to make sense of each method one line at a time. The following mathematical topics may be useful:

Pythagoras' theorem

Cosines of angles greater than $90^{\circ}$

Distance between two points on a coordinate grid

Trigonometry in right-angled triangles