There are lots of things to investigate. We hope you will find something interesting to contribute. Over the next few months we'll publish different bits and pieces and different methods until it adds up to a 'big picture' of this problem. Squares are constructed on the sides of a triangle and the outer vertices are joined as shown in the diagram. This is a problem that you might like to investigate using dynamic geometry software but whatever conjectures you are led to by the software will still need to be proved. (1) Prove that all four triangles have the same area. (For convenience in referring to the diagram label the triangles $D, r, s$ and $t$ and angles $A, B, C, x, y$ and $z$ as shown).