### 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?

### Bendy Quad

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.

# Hexy-metry

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

 A hexagon is inscribed inside a circle. The sides of the hexagon are alternately $a$ and $b$ units in length. What is the radius of the circle?