### 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.

# Xtra

### Why do this problem?

There are several different ways of solving this problem. It is a good example to bring home the point that, to be an expert problem solver, and to understand a piece of mathematics, it is not enough to find an answer. One should ask oneself "Have I used the best method?". This problem can be solved using trig formulae and the cosine rule but it can be solved more straightforwardly using only Pythagoras theorem.

### Possible approach

Challenge the class to find different methods of solution.

### Key question

How do we interpret the two solutions?

Is the diagram correct for both solutions?